{"action":"created","comment":{"url":"https://api.github.com/repos/petrelharp/stoch_proc_notes/pulls/comments/229416389","pull_request_review_id":169912868,"id":229416389,"node_id":"MDI0OlB1bGxSZXF1ZXN0UmV2aWV3Q29tbWVudDIyOTQxNjM4OQ==","diff_hunk":"@@ -0,0 +1,142 @@\n+\\documentclass[12pt]{article}\n+\n+\\usepackage[utf8]{inputenc} %For non-ASCII characters%\n+\\usepackage[margin=1in]{geometry} %For margins and related things%\n+\\usepackage{amssymb, amsthm, amsmath, bbm}\n+\n+\\let\\oldphi\\phi\n+\\renewcommand{\\phi}{\\varphi}\n+\n+\\newenvironment{soln}{\\begin{proof}[\\textsc{Solution}]}{\\renewcommand{\\qedsymbol}{$\\blacklozenge$}\\end{proof}}\n+\n+\\title{Poisson Point Processes Cont.}\n+\\author{Erin McCarthy\\thanks{\\textbf{email:} emccarth@uoregon.edu}}\n+\\date{\\today}\n+\\begin{document}\n+\\maketitle\n+\n+\\section{The Cauchy Distribution}\n+\n+$C \\sim Cauchy(1)$ \n+if $\\mathbb{P}\\{c \\in dx\\}= \\dfrac{dx}{\\pi(1+x)^2}$ and $t \\times C \\sim Cauchy(t)$\n+\n+\\noindent\\rule{4cm}{0.4pt} \n+\n+\\noindent$N \\sim PPP(\\dfrac{dtdx}{|x|^2})$ on $[0,\\infty) \\times \\{ \\mathbb{R}\\setminus \\{0\\}\\}$\n+\n+\\noindent$Let C_t = ($sum of x-values of points of $[0,t] \\times \\mathbb{R})= \\sum x \\mathbbm{1}_{[0,t)}(t) dN(t,x)$ \n+\n+\\subsection*{Claim}\n+$\\dfrac{1}{t} C_t \\stackrel{d}{=} C_1$\n+\n+\\begin{proof}\n+If $N = \\sum \\delta_{(t_i,x_i)} \\Rightarrow \\dfrac{1}{t} C_t = \\sum_{i:t_i\\leq1} \\dfrac{x_i}{t}$ but \\~N $= \\sum_i \\delta_{\\left(\\frac{t_i}{t_1}, \\frac{x_i}{t}\\right)} \\sim PPP(\\dfrac{dtdx}{x^2})$ i.e. $N \\stackrel{d}{=} $\\~N\n+(since $\\dfrac{dsdx}{x^2} = \\dfrac{tdu tdv}{t^2dv}=\\dfrac{dudv}{v^2}$, where $u=\\frac{s}{t}$ and $v=\\frac{x}{t}$), so $\\dfrac{1}{t}C_t = \\int x \\mathbbm{1}_{[0,1]}(t) d\\text{\\~{N}} (t,x) = \\text{\\~C}(1)$ \n+\\end{proof}\n+\n+\n+\\subsection*{Consequence}\n+Let $x_1,x_2,...,x_n$ iid $Cauchy(1)$ then $\\bar{x} = \\dfrac{1}{n} (x_1+...+x_n) \\sim Cauchy(1)$. This implies that that taking more samples does not tell you more about the mean! But ...\\\\\n+$$\\begin{aligned}\n+var[\\bar{x}] = \\frac{1}{n}var[x_1] = \\int_0^1 \\int_{-\\infty}^{\\infty} x^2 \\frac{dxdt}{x^2} = \\infty \\\\ \\text{ even } \\mathbb{E}[x_1]=\\int_0^1\\int_{-\\infty}^\\infty x \\frac{dxdt}{x^2} = \\text{ UND }\\\\\n+\\end{aligned}$$\n+\n+\n+\n+\\subsection*{Question:}\n+Suppose we have arrival times on $[0,\\infty).$ Are they (uniform) Poisson?\n+\n+Let $0< T_1 <T_2<..$ be the points, compare the distribution of $(T_{k+1}-T_k)^{N}_{k=1}$ inter-arrival intervals to the exponential distribution. e.g by Kolmogorov-Smirnov test (say that 10 times fast).\n+\n+We discussed two approaches to this problem in class including: looking the the length between the points or sectioning it into areas to examine the distribution.\n+\n+\\subsection*{Problem: London bombings} Let ${x_1,...,x_m}$ be points in the (2D) torus, T. Are they from a $PPP(m\\times\\lambda)$ i.e. constant intensity M?\\\\\n+\\textbf{Approach:} look for over-or under- dispersal.\n+\\begin{enumerate}\n+    \\item Estimate $m$: if $M \\sim Pois(m) \\Rightarrow \\text{then } M \\approx m$\n+    \n+    \\item Estimate local density: \n+    \n+    Let $\\rho(r):\\mathbb{R}_{\\geq 0}\\rightarrow \\mathbb{R}_{\\geq 0}$, where $\\rho(r)=0$ if $n>\\frac{1}{2}$, $\\rho(r) \\geq 0$, and $\\int_{T} \\rho(|x|)dx=1$.\n+    \n+    When $N=\\sum_i \\delta {x_i}$ and  $N\\sim PPP(m)$, let $P_i = \\sum\\limits_{j\\neq i} \\rho (|x_j-x_i|)$ and $P=\\sum_i P_i =\\int_{T\\times T\\setminus diag(T)} e^{(|x-y|)}N(dx)N(dy)$. Then:","path":"LectureNotes/Lectures/Fall2018/Lecture6/ASPlecturenotes.tex","position":62,"original_position":62,"commit_id":"9883cfd57fd925464bee5dd8eab21a7db552363f","original_commit_id":"9883cfd57fd925464bee5dd8eab21a7db552363f","user":{"login":"petrelharp","id":1046249,"node_id":"MDQ6VXNlcjEwNDYyNDk=","avatar_url":"https://avatars2.githubusercontent.com/u/1046249?v=4","gravatar_id":"","url":"https://api.github.com/users/petrelharp","html_url":"https://github.com/petrelharp","followers_url":"https://api.github.com/users/petrelharp/followers","following_url":"https://api.github.com/users/petrelharp/following{/other_user}","gists_url":"https://api.github.com/users/petrelharp/gists{/gist_id}","starred_url":"https://api.github.com/users/petrelharp/starred{/owner}{/repo}","subscriptions_url":"https://api.github.com/users/petrelharp/subscriptions","organizations_url":"https://api.github.com/users/petrelharp/orgs","repos_url":"https://api.github.com/users/petrelharp/repos","events_url":"https://api.github.com/users/petrelharp/events{/privacy}","received_events_url":"https://api.github.com/users/petrelharp/received_events","type":"User","site_admin":false},"body":"```suggestion\r\n    Let $\\{x_i\\}$ be the points of a PPP of constant intensity $m$, i.e., $N=\\sum_i \\delta_{x_i}$ with  $N\\sim PPP(m)$.\r\n    Then we use the weighting kernel $\\rho$ to count up the number of neighbors of each point: $P_i = \\sum\\limits_{j\\neq i} \\rho (|x_j-x_i|)$ and $P=\\sum_i P_i =\\int_{T\\times T\\setminus diag(T)} e^{(|x-y|)}N(dx)N(dy)$. Then:\r\n```","created_at":"2018-10-30T17:47:26Z","updated_at":"2018-10-30T17:47:27Z","html_url":"https://github.com/petrelharp/stoch_proc_notes/pull/5#discussion_r229416389","pull_request_url":"https://api.github.com/repos/petrelharp/stoch_proc_notes/pulls/5","author_association":"OWNER","_links":{"self":{"href":"https://api.github.com/repos/petrelharp/stoch_proc_notes/pulls/comments/229416389"},"html":{"href":"https://github.com/petrelharp/stoch_proc_notes/pull/5#discussion_r229416389"},"pull_request":{"href":"https://api.github.com/repos/petrelharp/stoch_proc_notes/pulls/5"}}},"pull_request":{"url":"https://api.github.com/repos/petrelharp/stoch_proc_notes/pulls/5","id":226313834,"node_id":"MDExOlB1bGxSZXF1ZXN0MjI2MzEzODM0","html_url":"https://github.com/petrelharp/stoch_proc_notes/pull/5","diff_url":"https://github.com/petrelharp/stoch_proc_notes/pull/5.diff","patch_url":"https://github.com/petrelharp/stoch_proc_notes/pull/5.patch","issue_url":"https://api.github.com/repos/petrelharp/stoch_proc_notes/issues/5","number":5,"state":"open","locked":false,"title":"Erins 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