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exact_match.csv

@@ -1,7 +1,7 @@
 Question,Answer,Domain,Difficulty,Type,Task,Reference,Explanation (Optional),,,
 "Consider a tripartite state <Math>\rho_{ABC}<\Math> consisting of d-dimensional qudits. Let A, B, C contain NA, NB, and NC qubits, respectively. Suppose there exist a channel R acting on the Hilbert space and outputing to the Hilbert space of AB such that <Math> || R[\rho_{BC}] - \rho_{ABC} ||_1 = \epsilon<\Math>. One can show that the conditional mutual information <Math>I(A:C|B) \le c*\sqrt{\epsilon}<\Math>. How does the prefactor c depend on d, NA, NB, and NC? You can throw away all the constants",<Math>d^NC<\Math>,Math/Physics,Easy,Advanced Reasoning,Quantum Information,Yifan Zhang,,,Quantum Information,15
 "Given a TFIM Hamiltonian <Math>H_L = \sum_{i=1}^{L-1} X_i X_{i+1} + \sum_{i=1}^L Z_i</Math> of size L, rewrite <Math>H_{L+1}</Math> as a function of <Math>H_L</Math>",H_{L+1} = H_L \otimes I + X_L X_{L+1} + Z_{L+1},Math/Physics,Easy,Advanced Reasoning,Condense Matter Physics,Roger,,,Condense Matter Physics,6
-"Given a TFIM Hamiltonian <Math>H_L = \sum_{i=1}^{L-1} X_i X_{i+1} + \sum_{i=1}^L Z_i</Math> of size L, rewrite <Math>H_{2L+2}</Math> as a function of <Math>H_L</Math> such that the expression splits the total system into a bipartite system connected by 2 sites, one is <Math>H_L^{sys}</Math> represents the system, the other is <Math>H_L^{env)</Math> represents the environment",H_{L+1} = H_L \otimes I_{L+2} + I_{L+2}\otimes H_L + X_L X_{L+1} + X_{L+1} X_{L+2} + X_{L+2}X_{L+3} + Z_{L+1} + Z{L+2},Math/Physics,Medium,Advanced Reasoning,Condense Matter Physics,Roger,,,Probability/Statsitics,1
+"Given a TFIM Hamiltonian <Math>H_L = \sum_{i=1}^{L-1} X_i X_{i+1} + \sum_{i=1}^L Z_i</Math> of size L, rewrite <Math>H_{2L+2}</Math> as a function of <Math>H_L</Math> such that the expression splits the total system into a bipartite system connected by 2 sites, one is <Math>H_L^{sys}</Math> represents the system, the other is <Math>H_L^{env)</Math> represents the environment",H_{L+1} = H_L \otimes I_{L+2} + I_{L+2}\otimes H_L + X_L X_{L+1} + X_{L+1} X_{L+2} + X_{L+2}X_{L+3} + Z_{L+1} + Z{L+2},Math/Physics,Medium,Advanced Reasoning,Condense Matter Physics,Roger,,,Probability/Statsitics,2
 "We can expand the power of the adjoint of the sum of operators <Math>sum_{i=1}^n A_i</Math> with an operator B into a power form, in short <Math>ad^k_{sum_{i=1}^n A_i} B = (?)^k</Math>",(sum_i ad_{A_i})^k (B),Math/Physics,Medium,Advanced Reasoning,Quantum Information,Roger,,,Astrophysics/Cosmology,5
 "Given a fermionic model <Math>H_L = \sum_{i=1}^{L-1} [a^{\dagger}_i a_{i+1} + a_i a^{\dagger}_{i+1} +n_i n{i+1}] </Math>, write it into a qubit format with Jordan Wigner transformation","H_L=\sum_{i=1}^{L-1}[\tfrac12\bigl(\sigma_i^{x}\sigma_{\,i+1}^{x}+\sigma_i^{y}\sigma_{\,i+1}^{y})+\tfrac14\bigl(\sigma_i^{z}\sigma_{\,i+1}^{z}+\sigma_i^{z}+\sigma_{\,i+1}^{z}+1)]",Math/Physics,Medium,Advanced Reasoning,Condense Matter Physics,Yuxuan Zhang,,,,0
 "Consider a bipartite state <Math>\rho_{AB}</Math> consisting of d-dimensional qudits. Let A, B contain NA, NB qubits, respectively. Find a bound on <Math>||\rho_{AB} - \rho_A \otimes I_B ||_1<\Math> that is a second moment quantity of <Math>\rho_{AB}<\Math>. I_B is the maximally mixed state on B.",<Math>d^{(NA+NB)/2} ||\rho_{AB} - \rho_A \otimes I_B ||_2<\Math>,Math/Physics,Medium,Advanced Reasoning,Quantum Information,Yifan Zhang,,,,0
@@ -83,30 +83,52 @@ the original state as |ψi⟩.",Math/Physics,Hard,Advanced Reasoning,Quantum Inf
 "How many different quantum states are there in a k-design? Assume the Hilbert space size is $d$, output your answer in combinatorics",<Math>\binom{d + k - 1}{k}<\Math>,Math/Physics,Easy,Basic Knowledge,Quantum Information,Yuxuan Zhang,,,,
 "For a spherical 3-design with N states, sample each quantum state according to the probability distribution of each state. What is the probability that a sampled state has a probability weight higher than 1/N? Hint: recall Paley–Zygmund inequality; you may assume the Hilbert space size d ->inf in the end",1/6,Math/Physics,Hard,Advanced Reasoning,Quantum Information,Yuxuan Zhang,,,,
 "T_n = \sum_i R_i 1(X_0>0), where R_i is the rank of X_i in the sorted order |X_1|\leq |X_2|.... Decompose T_n into a linear combination of U-statistic of order 1 and a U-statistic of order 2. (Hint: Express the result as aU_2+bU_2, where the coefficients a and b depend on n.)",\sum_{i<j} 1(X_1+X_j>0)+\sum_{i=i}^n 1(X_i>0) = \binom{n}{2}U_2 + nU_1,Math/Physics,Hard,Advanced Reasoning,Probability/Statsitics,Shunan Zheng,,,,
-"During the supernovae explosion, it will eject a large amount of material into the surrounding environment, with mass given by M_{ej}, and release amount of energy E_{SN}. Assume that during its evolution phase, the ionizing radiation has lower the ambient density of supernovae such that the explosion bubble could freely expand. Derive a rough estimate for the timescale of this exploding phase for a star with mass M_{\star}, assume all mass has been ejected away.","Assume the expansion halts when M_ej is equal to the mass swept away by the ejected mass: M_ej ~ M_swept = 4/3 pi r_{free expansion}^3 mu n_H m_p, where mu is the molecular weight of the universe. We derive the sweeping radius as r_{free expansion} ~(3M_{ej}/4 pi m_p n_H)^{1/3}. By the conservation of energy, the time scale for the expansion phase is t_{free expansion} ~  r_{free expansion} /(2E_{SN} / M_{ej})^{1/2} ~ M_{\star}^{5/6} (1/(4 pi m_p n_H))^{1/3}(1/ 2E_{SN})^{1/2}",Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,,,,
-"If we are living in a universe with primordial black holes as the only dark matter candidate (i.e. \Omega_{DM} = \Omega_{PBH}), and all of them are centered around some mass M_{PBH}, with number density given by n_{PBH}. What is the dominant change (Poisson fluctuation) induced by PBHs to the linear power spectrum P(k) related to the number density of PBHs n_{PBH}?","Assume PBHs are located at spatial coordinate \mathbf{x}, the overdense field at spatial point is given by: \delta_{iso}(\mathbf{x})\equiv \frac{\delta \rho_{PBH}}{\Bar{\rho}_{ DM}}=\frac{1}{n_{PBH}} \sum_i \delta_D\left(\mathbf{x}-\mathbf{x}_i\right)-1. Taking the two point correlation of the density fluctuation, we have \left\langle\delta_{PBH}(\mathbf{x}) \delta_{PBH}(\mathbf{0})\right\rangle =\frac{1}{n_{PBH}} \delta_D(\mathbf{x})+\xi_{PBH}(x). Taking the fourier transform of the twopoint correlation function, we have the power spectrum taking the form of P(k) = D_{0}^{2}/n_{ PBH} +.... where D_{0} is the growth factor. Therefore, we have P(k) \propto 1/n_{PBH}.
-",Math/Physics,Hard,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,,,,
-"Work out the temperature profile (T(r)) of the black hole accretion disk with radial denpendence (r), assume that black hole accretes at a rate \dot{M}, and assume that half of the graviational potential energy is converted into thermal heating energy, radiates away like black body","For a mass element M move from r+\delta r to r, the change in potential energy can be written as \delta E = GM_{BH}M \delta r / r^2. If half of the energy is converted to luminosity, we have \delta L = 2G M_{BH}\dot{M} / 2r^2 \delta r. If each segment of the accretion disk emits like a black body, the luminosity can be written as \delta L = 4 \pi r \delta r \sigma_{SB} T(r)^4. Solving for T(r), we have T(t) = (G M_{BH}\dot{M} / 8 \pi \sigma_{SB} r^3)^{1/4}",Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,,,,
-"In a standard LCDM universe, if a Black Hole seed is born in an ideal environment and always growing at an Eddington rate, with radiation efficiency of 0.1, what will be the mass of the BH at z =10 if it is born with a mass of 1000 Solar mass at z=20? (Give oder of magnitude estimate)","The characteristic Salpter time(e-fold growing time) scale for BH growth is 5\times 10^7 yr, where M(t) = M_{ini} exp(t/t_{Salpter}). Therefore, the time difference between redshift 20 and redshift 10 is about 3\times 10^8 yrs. Therefore, the BH mass estimated at z = 10 is about 4 \times 10^5 solar mass.",Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,,,,
-"Consider a dark matter halo at redshift $z$, capable of cooling through molecular hydrogen (H$\_2\$). The halo is characterized by its virial temperature $T_{\rm vir}$, molecular hydrogen cooling function $\Lambda(T, n_{\rm H})$, and hydrogen number density $n_{\rm H}$. To determine the minimum mass of such a halo capable of cooling within a local Hubble time, we impose the condition that the cooling time $\tau_{\rm cool}(M,z)$ be less than or equal to the local Hubble time $t_H(z)$. Using the equations provided below, derive the minimum halo mass $M_{t_H-\rm cool}(z)$.
+"During the supernovae explosion, it will eject a large amount of material into the surrounding environment (chracterized by number density n_H), with mass given by M_{ej}, and release amount of energy E_{SN}. Assume that during its evolution phase, the ionizing radiation has lower the ambient density of supernovae such that the explosion bubble could freely expand. Derive a rough estimate for the timescale of this exploding phase for a star with mass M_{\star}, assume all mass has been ejected away, using the quantities mentioned above.",t_{free expansion} ~  r_{free expansion} /(2E_{SN} / M_{ej})^{1/2} ~ M_{\star}^{5/6} (1/(4 pi m_p n_H))^{1/3}(1/ 2E_{SN})^{1/2},Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"Assume the expansion halts when M_ej is equal to the mass swept away by the ejected mass: M_ej ~ M_swept = 4/3 pi r_{free expansion}^3 mu n_H m_p, where mu is the molecular weight of the universe. We derive the sweeping radius as r_{free expansion} ~(3M_{ej}/4 pi m_p n_H)^{1/3}. By the conservation of energy, the time scale for the expansion phase is t_{free expansion} ~  r_{free expansion} /(2E_{SN} / M_{ej})^{1/2} ~ M_{\star}^{5/6} (1/(4 pi m_p n_H))^{1/3}(1/ 2E_{SN})^{1/2}",,,
+"If we are living in a universe with primordial black holes as the only dark matter candidate (i.e. \Omega_{DM} = \Omega_{PBH}), and all of them are centered around some mass M_{PBH}, with number density given by n_{PBH}. What is the dominant change (Poisson fluctuation) induced by PBHs to the linear power spectrum P(k) related to the number density of PBHs n_{PBH} (only show the proportional relationship)?"," P(k) \propto 1/n_{PBH}.
+",Math/Physics,Hard,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"Assume PBHs are located at spatial coordinate \mathbf{x}, the overdense field at spatial point is given by: \delta_{iso}(\mathbf{x})\equiv \frac{\delta \rho_{PBH}}{\Bar{\rho}_{ DM}}=\frac{1}{n_{PBH}} \sum_i \delta_D\left(\mathbf{x}-\mathbf{x}_i\right)-1. Taking the two point correlation of the density fluctuation, we have \left\langle\delta_{PBH}(\mathbf{x}) \delta_{PBH}(\mathbf{0})\right\rangle =\frac{1}{n_{PBH}} \delta_D(\mathbf{x})+\xi_{PBH}(x). Taking the fourier transform of the twopoint correlation function, we have the power spectrum taking the form of P(k) = D_{0}^{2}/n_{ PBH} +.... where D_{0} is the growth factor. Therefore, we have P(k) \propto 1/n_{PBH}.
+",,,
+"Work out the temperature profile (T(r)) of the black hole of mass M_{BH} accretion disk with radial denpendence (r), assume that black hole accretes at a rate \dot{M}, and assume that half of the graviational potential energy is converted into thermal heating energy, radiates away like black body. Express the answer in the quantities mentioned above.",T(t) = (G M_{BH}\dot{M} / 8 \pi \sigma_{SB} r^3)^{1/4},Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"For a mass element M move from r+\delta r to r, the change in potential energy can be written as \delta E = GM_{BH}M \delta r / r^2. If half of the energy is converted to luminosity, we have \delta L = 2G M_{BH}\dot{M} / 2r^2 \delta r. If each segment of the accretion disk emits like a black body, the luminosity can be written as \delta L = 4 \pi r \delta r \sigma_{SB} T(r)^4. Solving for T(r), we have T(t) = (G M_{BH}\dot{M} / 8 \pi \sigma_{SB} r^3)^{1/4}",,,
+"In a standard LCDM universe, if a Black Hole seed is born in an ideal environment and always growing at an Eddington rate, with radiation efficiency of 0.1, what will be the mass of the BH at z =10 if it is born with a mass of 1000 Solar mass at z=20? (Give order of magnitude estimate)",4 \times 10^5 solar mass.,Math/Physics,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"The characteristic Salpter time(e-fold growing time) scale for BH growth is 5\times 10^7 yr, where M(t) = M_{ini} exp(t/t_{Salpter}). Therefore, the time difference between redshift 20 and redshift 10 is about 3\times 10^8 yrs. Therefore, the BH mass estimated at z = 10 is about 4 \times 10^5 solar mass.",,,
+"Consider a dark matter halo at redshift $z$, capable of cooling through molecular hydrogen (H$\_2\$). The halo is characterized by its virial temperature $T_{\rm vir}$, molecular hydrogen cooling function $\Lambda(T, n_{\rm H})$, and hydrogen number density $n_{\rm H}$. To determine the minimum mass of such a halo capable of cooling within a local Hubble time, we impose the condition that the cooling time $\tau_{\rm cool}(M,z)$ be less than or equal to the local Hubble time $t_H(z)$. Using the equations provided below, derive the minimum halo mass $M_{t_H-\rm cool}(z)$ as a function of redshift z.
 
 \textbf{Given Equations related to cooling of hydrogen:}
-\tau\_{\rm cool}(M,z) = \frac{3 k\_B T\_{\rm vir}(M,z)}{2 \Lambda(T\_{\rm vir}, n\_{\rm H}) f\_{\rm H\_2}}, 
-\Lambda(T,n\_{\rm H}) \approx 10^{-31.6}\left(\frac{T}{100,\mathrm{K}}\right)^{3.4}\left(\frac{n\_{\rm H}}{10^{-4},\mathrm{cm}^{-3}}\right),\mathrm{erg,s^{-1}}, 
-f\_{\rm H\_2,max}(T) \approx 3.5 \times 10^{-4}\left(\frac{T}{1000,\mathrm{K}}\right)^{1.52}.
-","To find the minimum mass, we first equate the cooling time $\tau_{\rm cool}$ to the local Hubble time $t_H(z)$:
-\frac{3 k_B T_{\rm vir}(M,z)}{2 \Lambda(T_{\rm vir}, n_{\rm H}) f_{\rm H_2}} = t_H(z)
-Plugging in the expressions for $T_{\rm vir}$, $\Lambda$, and $n_{\rm H}$, we obtain:
-\tau_{\rm cool} \approx 3.46 \times 10^{10}\,\mathrm{s} \left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-5.4}f_{\rm H_2}^{-1}.
-Equating this to $t_H(z)$, we have:
-3.46 \times 10^{10}\,\mathrm{s} \left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-5.4}f_{\rm H_2}^{-1} = 3.216 \times 10^{15}\,\mathrm{s}\left(\frac{1+z}{31}\right)^{-3/2}.
-Solving for the required molecular hydrogen fraction:
-f_{\rm H_2} \approx 1.09 \times 10^{-5}\left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-3.9}.
-Next, we equate this required fraction to the maximum fraction that can form in the halo, $f_{\rm H_2,max}$, assuming $T=T_{\rm vir}$:
-1.09 \times 10^{-5}\left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-3.9} = 3.5 \times 10^{-4}\left(\frac{T_{\rm vir}}{1000\,\mathrm{K}}\right)^{1.52}.
-Replacing $T_{\rm vir}$ with its expression:
-T_{\rm vir}=2554\,\mathrm{K}\left(\frac{M}{10^6 M_\odot}\right)^{2/3}\left(\frac{1+z}{31}\right),
-we get:
-1.09 \times 10^{-5}\left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-3.9}=3.5\times 10^{-4}\left[2.554\left(\frac{M}{10^6 M_\odot}\right)^{2/3}\left(\frac{1+z}{31}\right)\right]^{1.52}.
-Solving for $M$, we obtain the minimum halo mass:
-M_{t_H-\rm cool} \approx 1.54 \times 10^{5} M_\odot \left(\frac{1+z}{31}\right)^{-2.074}.",Math/Physics,Hard,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,,,,
+\tau_{\rm cool}(M,z) = \frac{3 k_B T_{\rm vir}(M,z)}{2 \Lambda(T_{\rm vir}, n_{\rm H}) f_{\rm H_2}}, 
+T_{\rm vir}(M,z) &\approx 2554,\mathrm{K} \left(\frac{M}{10^6 M_\odot}\right)^{2/3} \left(\frac{1+z}{31}\right), 
+\Lambda(T,n_{\rm H}) \approx 10^{-31.6}\left(\frac{T}{100,\mathrm{K}}\right)^{3.4}\left(\frac{n_{\rm H}}{10^{-4},\mathrm{cm}^{-3}}\right),\mathrm{erg,s^{-1}}, 
+n_{\rm H} \approx 1.0\left(\frac{1+z}{31}\right)^3,\mathrm{cm}^{-3}, 
+t_H(z) \approx 3.216 \times 10^{15},\mathrm{s}\left(\frac{1+z}{31}\right)^{-3/2}, 
+f_{\rm H_2,max}(T) \approx 3.5 \times 10^{-4}\left(\frac{T}{1000,\mathrm{K}}\right)^{1.52}.
+", M_{t_H-\rm cool} \approx 1.54 \times 10^{5} M_\odot \left(\frac{1+z}{31}\right)^{-2.074}.,Math/Physics,Hard,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,"To find the minimum mass, we first equate the cooling time $\tau_{\rm cool}$ to the local Hubble time $t_H(z)$: \frac{3 k_B T_{\rm vir}(M,z)}{2 \Lambda(T_{\rm vir}, n_{\rm H}) f_{\rm H_2}} = t_H(z). Plugging in the expressions for $T_{\rm vir}$, $\Lambda$, and $n_{\rm H}$, we obtain: \tau_{\rm cool} \approx 3.46 \times 10^{10}\,\mathrm{s} \left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-5.4}f_{\rm H_2}^{-1}. Equating this to $t_H(z)$, we have: 3.46 \times 10^{10}\,\mathrm{s} \left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-5.4}f_{\rm H_2}^{-1} = 3.216 \times 10^{15}\,\mathrm{s}\left(\frac{1+z}{31}\right)^{-3/2}. Solving for the required molecular hydrogen fraction: f_{\rm H_2} \approx 1.09 \times 10^{-5}\left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-3.9}. Next, we equate this required fraction to the maximum fraction that can form in the halo, $f_{\rm H_2,max}$, assuming $T=T_{\rm vir}$: 1.09 \times 10^{-5}\left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-3.9} = 3.5 \times 10^{-4}\left(\frac{T_{\rm vir}}{1000\,\mathrm{K}}\right)^{1.52}. Replacing $T_{\rm vir}$ with its expression: T_{\rm vir}=2554\,\mathrm{K}\left(\frac{M}{10^6 M_\odot}\right)^{2/3}\left(\frac{1+z}{31}\right), we get: 1.09 \times 10^{-5}\left(\frac{M}{10^6 M_\odot}\right)^{-1.6}\left(\frac{1+z}{31}\right)^{-3.9}=3.5\times 10^{-4}\left[2.554\left(\frac{M}{10^6 M_\odot}\right)^{2/3}\left(\frac{1+z}{31}\right)\right]^{1.52}. Solving for $M$, we obtain the minimum halo mass: M_{t_H-\rm cool} \approx 1.54 \times 10^{5} M_\odot \left(\frac{1+z}{31}\right)^{-2.074}.",,,
+"The lifetime of a lightbulb is an exp(\lambda) random variable. Replacement happens every fixed T time unit, starting at time 0. What is the expected time when the lightbulb is discovered to have failed. Denote the time the lightbulb has failed with D.","
+E[D] = \frac{T}{1-e^{-\lambda T}}
+",Math/Physics,Medium,Advanced Reasoning,Probability/Statsitics,Shunan Zheng,,,,
+"The lifetime of a lightbulb follows exp(1/\theta), where the expected lifetime \theta is unknown. In one experiment, test N lightbulbs and record their exact lifetime y_1,...,y_N. In another experiment, test M lightbulbs.  At some time t, record if the lightbulbs are still burning with indicators Z_1,...,Z_M. Z_i = 1 if the lightbulb i is still burning; Z_i = 0 if the lightbulb i is out. To estimate the unknown \theta, implement the EM algorithm. What’s the corresponding E-step and M-step? (Hint: Denote the observed data with:  $y= (y_1, \ldots, y_N, Z_1, \ldots, Z_M)$. Let $Z = \sum_{i=1}^M Z_i $. Denote the unobserved data with: $X = (X_1, \ldots, X_M)$.)","
+The E-step:
+
+Q(\theta | \theta_t) 
+= E [l^c(\theta; y, X)] = -N ( \ln \theta + \bar{y}/\theta) - \sum_{i=1}^{M} ( \ln \theta + E(X_i | E_i)/\theta) 
+= -(N + M) \ln \theta - \frac{1}{\theta} ( N \bar{y} + Z(t + \theta_t) + (M - Z) ( \theta_t - \frac{t \exp(-t/\theta_t)}{1 - \exp(-t/\theta_t)} ) ).
+
+
+The M-step:
+
+\theta_{t+1} 
+=\arg\max_\theta Q(\theta \mid \theta_t) 
+= \frac{1}{N + M} ( \bar{y} + Z(t + \theta_t) + (M - Z) ( \theta_t - \frac{t \exp(-t/\theta_t)}{1 - \exp(-t/\theta_t)} ) ).
+
+",Math/Physics,Hard,Advanced Reasoning,Probability/Statsitics,Shunan Zheng,,,,
+"Let ${N(t), t \ geq 0}$ be an RP with integer valued inter-renewal times with pmf 
+$P(X_n = 0) = 1-\alpha$, $P(X_n = 1) = \alpha$, $n \geq 1$, where $0<\alpha<1$. Compute $P(N(t) = k)$ for k = 0,1,2,....
+",Math/Physics,Medium,Advanced Reasoning,Probability/Statsitics,Shunan Zheng,,,,
+"Assume the distribution of the population G_i  is:  $G_i(m = 1) \sim N(\mu^{(i)}, \sigma_i^2), \quad (i = 1, 2)$. According to the distance-based classification rule (assume $\mu^{(1)} > \mu^{(2)}$, $\sigma_1 < \sigma_2$):
+\[
+\begin{cases}
+x \in G_1, & \text{if } \mu^* < x < \mu^{(1)} \\
+x \in G_2, & \text{if } x \leq \mu^* \text{ or } x \geq \mu^{(1)}
+\end{cases}
+\]
+where  $\mu^* = \frac{\sigma_1 \mu^{(2)} + \sigma_2 \mu^{(1)}}{\sigma_1 + \sigma_2}$.
+Find the misclassification probabilities P(2|1).
+
+
+

multiple_choice.csv

@@ -28,8 +28,8 @@ which quantity cannot increase under local operations and classical communicatio
 "Say we have an n-qubit, equal bipartite quantum system. Gates within each bipartition is considered free. What is the lower bound of two-qubit gates required to implement an arbitrary unitary operation on the whole system? ",2^n,2^(n/2),2^(n/3),n^2,B,Hard,Advanced Reasoning,Quantum Information,Yuxuan Zhang,,Math/Physics,,,
 consider the hidden group problem of the groups below. giving a polynomal-time quantum algorithm to which one will lead to nontrivial complexity-theoretic consequences,order-n cyclic group,order-n dihedral group,order-n symmetric group,direct product of two order-n cyclic groups,BC,Medium,Basic Knowledge,Quantum Information,Yifan Zhang,,Math/Physics,,,
 Which of the following systems cannot be non-Hermitian?,random circuit with projected measurement,Ising model with imaginary external field,transmon coupled to a bosonic thermal bath,Optical system with photon loss,C,Medium,Advanced Reasoning,Quantum Information,Yuxuan Zhang,,Math/Physics,,,
-Why static single assignment is useful for representing quantum programs?,naturally embeds linear logic,allow dataflow analysis when value semantics are used for representing qubits,prevents user from defining program that violates non-cloning theorem,easy to express graph based representation like ZX diagrams,"A,B,C",Medium,Basic Knowledge,Computational Physics,Xiu-Zhe Luo,,Math/Physics,,,
-"Consider neutral atoms that allows using an optical tweezer to move the atoms, why this could have advantage when implementing deep circuits?",long coherent time,can create all-to-all connection via blockade effects,routing the atom moves is simpler than superconducting circuits,fidelity is higher than superconducting circuits,"A,B",Easy,Basic Knowledge,Quantum Information,Yuxuan Zhang,,Math/Physics,,,
+Why static single assignment is useful for representing quantum programs?,naturally embeds linear logic,allow dataflow analysis when value semantics are used for representing qubits,prevents user from defining program that violates non-cloning theorem,easy to express graph based representation like ZX diagrams,ABC,Medium,Basic Knowledge,Computational Physics,Xiu-Zhe Luo,,Math/Physics,,,
+"Consider neutral atoms that allows using an optical tweezer to move the atoms, why this could have advantage when implementing deep circuits?",long coherent time,can create all-to-all connection via blockade effects,routing the atom moves is simpler than superconducting circuits,fidelity is higher than superconducting circuits,AB,Easy,Basic Knowledge,Quantum Information,Yuxuan Zhang,,Math/Physics,,,
 GKP coding 1% encoding error corresponds to roughly how much squeezing? Choose the closest one,2dB,6dB,10dB,20dB,B,Medium,Basic Knowledge,Quantum Information,Yuxuan Zhang,,Math/Physics,,,
 Dropping a watermelon from the highest mountain on the moon. How long does it take to reach earth? choose the closest estimation,1h,10h,100h,10000h,D,Medium,Basic Knowledge,Quantum Information,Yuxuan Zhang,,Math/Physics,,,
 How many gates are required to distinguish a Haar random state from a maximally mixed state?,constant,log(n),poly(n),exp(n),D,Medium,Advanced Reasoning,Quantum Information,Yuxuan Zhang,,Math/Physics,,,
@@ -64,28 +64,28 @@ Why light have only 2 polarizations in quantum field theory?,photon is a ferimon
 Do photons interacts with itselves in quantum field theory? Why or why not?,photons interact with itselves because it is allowed by the rules of feymann diagram,"photons are force carrier particles, which do not interact with itselves",there is no photon-photon 3-way or 4-way interaction nodes in the Feymann diagrams,"photon-photon coupling is of the 4^th order in perturbation, the cross-section is too low",CD,Medium,Basic Knowledge,High-energy Physics,,,Math/Physics,,,
 What does the Eddington limit of the lumiosity means in Astronomy?,"The inward gravitational force
 balances the radiation force induced
-by electron scattering",The luminosity of the star is stronger than the flux of a black hole of the same mass,The luminosity of a star when the star is leaving the main branch phase in the HR diagram,"The theoretical upper limit of luminosity when the emitting object (star, BH accretion disk, etc.) is in hydrostatic equilibrium.","A, D",Medium,Basic Knowledge,High-energy Physics,,,Math/Physics,,,
-Which types of observation can astronomers conduct currently which allow us to get some constraints of the spin of a black hole,Observing the black hole event horizon with visble light ,"The observation of a powerful, relativistic jet produced from a black hole indicates a relatively large spin (and strong magnetic field)",Fe Kalpha fluorescent lines observed from AGNs and X-ray binaries.,The gravitational wave signiture of black holes binaries,"B, C, D",Hard,Advanced Reasoning,High-energy Physics,,,Math/Physics,,,
+by electron scattering",The luminosity of the star is stronger than the flux of a black hole of the same mass,The luminosity of a star when the star is leaving the main branch phase in the HR diagram,"The theoretical upper limit of luminosity when the emitting object (star, BH accretion disk, etc.) is in hydrostatic equilibrium.",AD,Medium,Basic Knowledge,High-energy Physics,,,Math/Physics,,,
+Which types of observation can astronomers conduct currently which allow us to get some constraints of the spin of a black hole,Observing the black hole event horizon with visble light ,"The observation of a powerful, relativistic jet produced from a black hole indicates a relatively large spin (and strong magnetic field)",Fe Kalpha fluorescent lines observed from AGNs and X-ray binaries.,The gravitational wave signiture of black holes binaries,BCD,Hard,Advanced Reasoning,High-energy Physics,,,Math/Physics,,,
 "In a two dimensional space/spacetime, the metric tensor is given by g_{xy}=g_{yx} = 1/y, and g_{xx}=g_{yy}=0, is there any physical significance for the singularity y = 0?","Yes, because at y = 0, the metric tensor diverges.","No, because at y = 0, diverging terms are allowed in metric tensor in General Relativity","Yes, because for a physically resonable metric tensor, the diagonal terms are non-zero","No, because we can choose another coordinates, such that the singularity at y = 0 vanishes, so the singularity is a coordinate singularity, but not a physical singularity.",D,Hard,Advanced Reasoning,High-energy Physics,,,Math/Physics,,,
-What is the necessity of renormalization in Quantum Field Theory?,"Renormalization are not necessary when calculating the interaction corss section, because the tree-level diagram do not have unconstrained freedom.","Feynman diagram with at least 1 loops have a unconstrained degree of freedom, which makes the cross section infinite","When calculating the intersection cross section, we need to sum over the effect of all possible feynman diagrams, which requires a treatment of the infinities",Renormalization is well-defined mathematically and it helps the cross section calculation to be mathematically rigorous.,"B, C",Medium,Basic Knowledge,High-energy Physics,,,Math/Physics,,,
+What is the necessity of renormalization in Quantum Field Theory?,"Renormalization are not necessary when calculating the interaction corss section, because the tree-level diagram do not have unconstrained freedom.","Feynman diagram with at least 1 loops have a unconstrained degree of freedom, which makes the cross section infinite","When calculating the intersection cross section, we need to sum over the effect of all possible feynman diagrams, which requires a treatment of the infinities",Renormalization is well-defined mathematically and it helps the cross section calculation to be mathematically rigorous.,BC,Medium,Basic Knowledge,High-energy Physics,,,Math/Physics,,,
 "In General Relativity, the twin paradox is commonly discussed. Suppose one of the twin takes a spaceship travels far and be back, and the other twin waited on Earth, which twin will be older.",The twin who took the spaceship,The twin who stayed on Earth,They are equally old,The information provided is insufficient,B,Easy,Basic Knowledge,High-energy Physics,,,Math/Physics,,,
-"Quantum teleportation is very useful in quantum communication and distributed quantum computation, which of the following are correct?",Quantum teleportation teleports quantum-size particles described by wave function,Quantum teleportation requires pre-shared entanglements,States can be teleported instantly with no restriction on distance,Classical communication and post processing is needed,"B, D",Easy,Basic Knowledge,Computational Physics,,,Math/Physics,,,
+"Quantum teleportation is very useful in quantum communication and distributed quantum computation, which of the following are correct?",Quantum teleportation teleports quantum-size particles described by wave function,Quantum teleportation requires pre-shared entanglements,States can be teleported instantly with no restriction on distance,Classical communication and post processing is needed,BD,Easy,Basic Knowledge,Computational Physics,,,Math/Physics,,,
 "Generalizing the idea of topological quantum error correction from Torus to Klein bottles and 2D real projective plane. How many logical qubits can be encoded by Torus, Klein bottles and 2D real projective plane respectively?","2, 2, 1 qubit","2, 2, 2 qubit","2, 1, 1 qubit","2, 1, 2 qubit",A,Hard,Advanced Reasoning,Computational Physics,,,Math/Physics,,,
 Which of the following represents a complete gate set for Universal quantum computation? ,"X, Y, Z, CNOT, P","X, Z, CNOT, H, P","T, X, Y, H, P","T, Z, CNOT, H, P",D,Medium,Advanced Reasoning,Computational Physics,,,Math/Physics,,,
 "On a topological quantum error correction surface code on a L-by-L lattice (so 2L^2 physical qubits), if the physical error rate is about half of the threshold error rate, and when L=7, the logical error rate is about 10^{-3}. Approximately how many physical qubits are needed to lower the logical error rate to 10^{-10}?",50,500,5000,50000,C,Hard,Advanced Reasoning,Computational Physics,,,Math/Physics,,,
-"In order to observe Hong–Ou–Mandel effect (input modes are a and b, output modes are c and d), which of the following conditions are required?",the photons arrive at the beam-splitter at the same time,the input modes should be single photons of the same frequency,the input modes are entangled,A polarizing beam splitter should be used,"A, B",Medium,Advanced Reasoning,Condensed Matter Physics,,,Math/Physics,,,
-Shor's factorization algorithm belongs to which of the following complexity classes?,NP,BQP,NP-complete,BPP,"A, B",Easy,Basic Knowledge,Computational Physics,Xiu-Zhe Luo,,Math/Physics,,,
-Which of the following statments about quantum bit commitment are correct?,"Bit commitment is impossible classically, but posible with quantum computer, because measurement will collapse the quantum state","Quantum bit commitment is possible in theory, but unrealistic with modern technology","Quantum bit commitment is impossible in theory, because commitment can always be changed with local operations",The impossibility of quantum bit commitment lies in the schmidt decomposition,"C, D",Medium,Basic Knowledge,Computational Physics,Xiu-Zhe Luo,,Math/Physics,,,
+"In order to observe Hong–Ou–Mandel effect (input modes are a and b, output modes are c and d), which of the following conditions are required?",the photons arrive at the beam-splitter at the same time,the input modes should be single photons of the same frequency,the input modes are entangled,A polarizing beam splitter should be used,AB,Medium,Advanced Reasoning,Condensed Matter Physics,,,Math/Physics,,,
+Shor's factorization algorithm belongs to which of the following complexity classes?,NP,BQP,NP-complete,BPP,AB,Easy,Basic Knowledge,Computational Physics,Xiu-Zhe Luo,,Math/Physics,,,
+Which of the following statments about quantum bit commitment are correct?,"Bit commitment is impossible classically, but posible with quantum computer, because measurement will collapse the quantum state","Quantum bit commitment is possible in theory, but unrealistic with modern technology","Quantum bit commitment is impossible in theory, because commitment can always be changed with local operations",The impossibility of quantum bit commitment lies in the schmidt decomposition,CD,Medium,Basic Knowledge,Computational Physics,Xiu-Zhe Luo,,Math/Physics,,,
 "In a (1+1)D conformal field theory (CFT), the Rényi entropy Sn​(ℓ) of a subsystem of length ℓ at zero temperature behaves as",Sn​(ℓ)∼ℓ^n,Sn​(ℓ)∼logℓ,Sn​(ℓ)∼ℓlogℓ,Sn​(ℓ)∼ℓ^(1/n),B,Medium,Basic Knowledge,High-energy Physics,Di Luo,,Math/Physics,,,
 "In the BCS theory of superconductivity, which of the following best describes the nature of the Cooper pairs at zero temperature?",Real-space localized bosons with strong repulsion,Pairs of electrons bound by Coulomb attraction,Momentum-space correlated states with opposite momenta and spin,Entangled electron-hole pairs near the Fermi surface,C,Easy,Basic Knowledge,Condensed Matter Physics,Di Luo,,Math/Physics,,,
-Which of the following statements are true regarding the entanglement entropy in (1+1)D critical systems described by conformal field theory (CFT)?,The von Neumann entropy scales logarithmically with subsystem size.,The coefficient of the log term is proportional to the central charge.,The entanglement entropy is invariant under conformal transformations.,The Rényi entropy becomes independent of nn in the thermodynamic limit.,"A,B",Medium,Basic Knowledge,High-energy Physics,Di Luo,,Math/Physics,,,
-Which of the following are necessary conditions to violate the Bell-CHSH inequality in a quantum experiment?,A maximally entangled bipartite pure state,Measurement settings that are incompatible observables,Use of a quantum channel with nonzero mutual information,Postselection of coincident events in detection,"A,B",Easy,Advanced Reasoning,Quantum Information,Di Luo,,Math/Physics,,,
-Which of the following statements about quantum error-correcting codes are true?,CSS codes require the stabilizer group to be abelian.,Transversal gates always commute with stabilizers.,The code distance determines the number of detectable errors.,Logical operators must commute with all stabilizers but not necessarily with each other.,"A,C,D",Medium,Advanced Reasoning,Quantum Information,Di Luo,,Math/Physics,,,
-Which of the following are correct interpretations of the Berry curvature in the Brillouin zone of a 2D Bloch band?,It contributes to the Hall conductivity via the TKNN formula.,It is always zero for time-reversal symmetric systems.,It defines a fictitious magnetic field in momentum space.,Its integral over the Brillouin zone gives the Chern number.,"A,C,D",Easy,Basic Knowledge,Condensed Matter Physics,Di Luo,,Math/Physics,,,
-Which statements are true regarding confinement in compact U(1) gauge theory in 2+1 dimensions?,Instantons lead to confinement of electric charges,The Wilson loop obeys a area law in the confined phase,The presence of a Chern-Simons term preserves confinement,Magnetic monopoles proliferate and gap the photon mode,"A,B,D",Medium,Basic Knowledge,High-energy Physics,Di Luo,,Math/Physics,,,
-Which statements about spontaneous symmetry breaking (SSB) in quantum field theory are correct?,"In 1+1D, continuous symmetries cannot be spontaneously broken at finite temperature.",Goldstone’s theorem applies only to continuous global symmetries.,Discrete symmetry breaking in 1+1D is forbidden at any temperature.,"In 2+1D, continuous symmetries can be spontaneously broken, and Goldstone bosons can appear.","A,B,D",Easy,Advanced Reasoning,High-energy Physics,Di Luo,,Math/Physics,,,
+Which of the following statements are true regarding the entanglement entropy in (1+1)D critical systems described by conformal field theory (CFT)?,The von Neumann entropy scales logarithmically with subsystem size.,The coefficient of the log term is proportional to the central charge.,The entanglement entropy is invariant under conformal transformations.,The Rényi entropy becomes independent of nn in the thermodynamic limit.,AB,Medium,Basic Knowledge,High-energy Physics,Di Luo,,Math/Physics,,,
+Which of the following are necessary conditions to violate the Bell-CHSH inequality in a quantum experiment?,A maximally entangled bipartite pure state,Measurement settings that are incompatible observables,Use of a quantum channel with nonzero mutual information,Postselection of coincident events in detection,AB,Easy,Advanced Reasoning,Quantum Information,Di Luo,,Math/Physics,,,
+Which of the following statements about quantum error-correcting codes are true?,CSS codes require the stabilizer group to be abelian.,Transversal gates always commute with stabilizers.,The code distance determines the number of detectable errors.,Logical operators must commute with all stabilizers but not necessarily with each other.,ACD,Medium,Advanced Reasoning,Quantum Information,Di Luo,,Math/Physics,,,
+Which of the following are correct interpretations of the Berry curvature in the Brillouin zone of a 2D Bloch band?,It contributes to the Hall conductivity via the TKNN formula.,It is always zero for time-reversal symmetric systems.,It defines a fictitious magnetic field in momentum space.,Its integral over the Brillouin zone gives the Chern number.,ACD,Easy,Basic Knowledge,Condensed Matter Physics,Di Luo,,Math/Physics,,,
+Which statements are true regarding confinement in compact U(1) gauge theory in 2+1 dimensions?,Instantons lead to confinement of electric charges,The Wilson loop obeys a area law in the confined phase,The presence of a Chern-Simons term preserves confinement,Magnetic monopoles proliferate and gap the photon mode,ABD,Medium,Basic Knowledge,High-energy Physics,Di Luo,,Math/Physics,,,
+Which statements about spontaneous symmetry breaking (SSB) in quantum field theory are correct?,"In 1+1D, continuous symmetries cannot be spontaneously broken at finite temperature.",Goldstone’s theorem applies only to continuous global symmetries.,Discrete symmetry breaking in 1+1D is forbidden at any temperature.,"In 2+1D, continuous symmetries can be spontaneously broken, and Goldstone bosons can appear.",ABD,Easy,Advanced Reasoning,High-energy Physics,Di Luo,,Math/Physics,,,
 "How many disconnected components does the spacetime symmetry of 1+1d spin-1/2 theory, Spin(1,1), have?",1,2,4,8,C,Medium,Basic Knowledge,High-energy Physics,Wucheng Zhang,,Math/Physics,,,
-Which of the followings are redundant operators in 1+1d Majorana conformal field theory?,<Math>i\chi_\text{L}\partial_x\chi_\text{L}+i\chi_\text{R}\partial_x\chi_\text{R}<\Math>,<Math>i\chi_\text{L}\partial_x\chi_\text{R}<\Math>,<Math>i\chi_\text{L}\partial_x\chi_\text{L}<\Math>,<Math>i\chi_\text{L}\partial_x^2\chi_\text{R}<\Math>,"B,C,D",Hard,Advanced Reasoning,High-energy Physics,Wucheng Zhang,,Math/Physics,,,
+Which of the followings are redundant operators in 1+1d Majorana conformal field theory?,<Math>i\chi_\text{L}\partial_x\chi_\text{L}+i\chi_\text{R}\partial_x\chi_\text{R}<\Math>,<Math>i\chi_\text{L}\partial_x\chi_\text{R}<\Math>,<Math>i\chi_\text{L}\partial_x\chi_\text{L}<\Math>,<Math>i\chi_\text{L}\partial_x^2\chi_\text{R}<\Math>,BCD,Hard,Advanced Reasoning,High-energy Physics,Wucheng Zhang,,Math/Physics,,,
 What is the order of anomaly of time-reversal symmetry and fermion parity in quantum mechanics?,2,4,8,16,C,Easy,Basic Knowledge,High-energy Physics,Wucheng Zhang,,Math/Physics,,,
 "In 3+1 fermionic theory in the continuum, assuming Lorentz symmetry, what is the square of CRT and CPT? (R is the reflection of one space coordinate and P is the reflection of all space coordinate)?","1,1","1,(-1)^F","(-1)^F,1","(-1)^F,(-1)^F",B,Medium,Advanced Reasoning,High-energy Physics,Wucheng Zhang,,Math/Physics,,,
 Which of the following is true about GSO projection?,It is a construction of modular-invariant string thoery,It sums over R and NS boundary conditions on the fermion fields and projects onto states of definite world-sheet fermion number.,"In supersymmetric string thoeries, there are independent GSO projections on left-movers and right-movers. ",The diagnoal prjection produces supersymmetric thoery.,ABC,Easy,Basic Knowledge,High-energy Physics,Wucheng Zhang,,Math/Physics,,,
@@ -119,4 +119,14 @@ Which of the following statement is true about the Jeans instability?,Jeans radi
 Which of the following statement is true about the Toomre's stability criterion?,"The faster the gas disk rotates, the more unstable it is against collapse","For star clusters, if the disk contain a dense surface number density of stars, the less likely it will collapse",It can be applied to both gaseous/stellar disks,"If the value Q is less than 1, the disk will be unstable against collapse",CD,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,,Math/Physics,,,
 "If a stellar system of mass M has a median mass of r_h reaches virial equilibrium, what is the cloest expression for the root mean square of velocity of stars within this system?",GM^2/0.4 r_h,GM/0.4 r_h,GM/ r_h^2,4GM/3 r_h,B,Easy,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,,Math/Physics,,,
 "What is the cloest number for the surface density of the stars near the sun, if sun is orbiting around milky way at 220 km^-1 locating at 8.5 kpc from the center?",1 Msun /pc^-2,20 Msun /pc^-2,200 Msun /pc^-2,20000 Msun /pc^-2,A,Medium,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,,Math/Physics,,,
-Which of the following is true about the luminosity function found by Schecter's Law?,There is no tunable parameters within the equation,The function implies a large number of galaxies with low luminosities as compared to more luminous ones,This law holds for galaxies of all luminsities and of all kinds,It can also be written in magnitude system instead of luminosity,BD,Hard,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,,Math/Physics,,,
+Which of the following is true about the luminosity function found by Schecter's Law?,There is no tunable parameters within the equation,The function implies a large number of galaxies with low luminosities as compared to more luminous ones,This law holds for galaxies of all luminsities and of all kinds,It can also be written in magnitude system instead of luminosity,BD,Easy,Advanced Reasoning,Astrophysics/Cosmology,Saiyang Zhang,,Math/Physics,,,
+Which of the following metrics is a special case of p-Wasserstein metric when p=2?,Kantorovich,Lévy–Fréchet,Kullback Liebler,Pearson chi-squared,A,Hard,Basic Knowledge,Probability/Statistics,,,,,,
+Which of the following normality tests does not require the exact empirical distribution of the samples?,Kolmogorov–Smirnov, Chi-Square Test,Anderson–Darling,Lilliefors,B,Medium,Advanced Reasoning,Probability/Statistics,,,,,,
+"For which distribution is the beta distribution a conjugate prior?
+"For normal distribution with random variance \sigma^2, fixed mean \mu, what is its conjugate prior?",Gamma,IG(Inverse Gamma),Gumbel,Weibul,B,Medium,Basic Knowledge,Probability/Statistics,,,,,,
+What function is Hellinger divergence based on?,(t-1)log(t),(t-1)^2,(\sqrt{t}-1)^2,t log(t),C,Easy,Basic Knowledge,Probability/Statistics,,,,,,
+"A random variable $X~F(n,m)$, what distribution does $Z =\frac{(n/m)X}{1+(n/m)X}$ follow?","Beta(m,n)","Beta(n/2,m/2)","F(n+1,m)",Power(m+n),B,Medium,Advanced Reasoning,Probability/Statistics,,,,,,
+"Randomly sample x_1,...,x_12 from $N~N(0,\sigma^2)$, what’s the distribution of $y = \frac{\sum_{i=1}^{8} x_i}{2\sum_{i=9}^{12} x_i}$?","F(4,8)","F(8,4)","Beta(4,4)","Beta(4,8)",B,Easy,Advanced Reasoning,Probability/Statistics,,,,,,
+Which of the following statements about Little’s Law is false?,It is derived by averaging the total customer time over a sample path,It requires exponential service times.,It assumes the system is in steady state.,It requires every arriving customer to eventually depart.,B,Easy,Basic Knowledge,Probability/Statistics,,,,,,
+Which of the following is not a valid way to define a Poisson process?,"It has independent, stationary increments and $N(t) ~ Poisson(\lambda t)$",The interarrival times are i.i.d. exponential with rate \lambda,It is the limit of Bernoulli trials with success probability \lambda \Delta t per interval,"The number of arrivals in time t is Binomial(n, \lambda t/n)",D,Medium,Basic Knowledge,Probability/Statistics,,,,,,
+"Let $X_1, ..., X_n ~ Pareto(\alpha, k)$ i.i.d. What is the limiting distribution of $M_n = \max{X_1, ..., X_n}$ as n approaches infinity?",\alpha e^{-\alpha x},\frac{1}{\alpha}e^{\frac{x}{\alpha}+e^{\frac{x}{\alpha}}},e^{-x^{\alpha}},ke^{-kx},C,Hard,Advanced Reasoning,Probability/Statistics,,,,,,