<s>[INST] The derivative of the cumulative distribution of $\min(X_1, a_1) + \min(X_2, a_2)$ Assume $X_1, X_2$ are continuous random variables in $L^{\infty}$. Let $a_1, a_2$ be real numbers and
$$Y := \min(X_1, a_1)+ \min(X_2, a_2).$$
The dependency structure between $X_1$ and $ X_2$ is not known, but we assume that the joint and marginal distributions are well-defined.
Is there a way to calculate
\begin{align}
\frac{d}{da_i} F_{Y}(t), 
\end{align}
where $F_{Y}(t) := P(\min(X_1, a_1)+ \min(X_2, a_2) \leq t)$?
I have tried to find an explicit expression for $Y$, with the idea to express the cumulative distribution function as a double integral, with the hope of appealing to the fundamental theorem of calculus, but I cannot seem to find an expression that makes sense.
 [/INST] We can write, for  the first term
$$
\eqalign{
  & \Pr \left( {\min \left( {X_{\,1} ,a_{\,1} } \right) \le t_{\,1} } \right)
 = F_{\,m} \left( {t_{\,1} ;\,a_{\,1} } \right) =   \cr 
  &  = \left[ {t_{\,1}  < a_{\,1} } \right]\Pr \left( {X_{\,1}  \le t_{\,1} } \right)
 + \left[ {a_{\,1}  \le t_{\,1} } \right]1 =   \cr 
  &  = \left( {1 - \left[ {a_{\,1}  \le t_{\,1} } \right]} \right)\Pr \left( {X_{\,1}  \le t_{\,1} } \right)
 + \left[ {a_{\,1}  \le t_{\,1} } \right] =   \cr 
  &  = \Pr \left( {X_{\,1}  \le t_{\,1} } \right)
 + \left[ {a_{\,1}  \le t_{\,1} } \right]\left( {1 - \Pr \left( {X_{\,1}  \le t_{\,1} } \right)} \right) =   \cr 
  &  = \left[ {t_{\,1}  < a_{\,1} } \right]F_{\,1} \left( {t_{\,1} } \right)
 + \left[ {a_{\,1}  \le t_{\,1} } \right] =   \cr 
  &  = F_{\,1} \left( {t_{\,1} } \right)
 + \left[ {a_{\,1}  \le t_{\,1} } \right]\left( {1 - F_{\,1} \left( {t_{\,1} } \right)} \right) =   \cr 
  &  = F_{\,1} \left( {t_{\,1} } \right)
 + H\left( {t_{\,1}  - a_{\,1} } \right)\left( {1 - F_{\,1} \left( {t_{\,1} } \right)} \right) \cr} 
$$
where

*

*the square brackets denote the Iverson bracket ;

*H is the Heaviside step function ($H(0)=1$);
and where we consider $a_1$ as a parameter.

Analogously for the other term in $X_2$.
Now, for computing the CDF of the sum of  two terms we need  the PDF of at least one of the distributions,
i.e. the derivative of the expression above.
This shall be done with caution, considering that there is a finite jump at $t_1 = a_1$.
Only at that discontinuity we are using the Heaviside function, and its formal derivative as a Dirac Delta.
Thus we write
$$
\eqalign{
  & P\left( {\min \left( {X_{\,1} ,a_{\,1} } \right) = t_{\,1} } \right)
 = f_{\,m} \left( {t_{\,1} ;\,a_{\,1} } \right) =   \cr 
  &  = {d \over {dt_{\,1} }}F_{\,m} \left( {t_{\,1} ;\,a_{\,1} } \right) =   \cr 
  &  = \left[ {t_{\,1}  < a_{\,1} } \right]f_{\,1} \left( {t_{\,1} } \right)
 + \left( {1 - F_{\,1} \left( {a_{\,1} } \right)} \right)\delta \left( {t_{\,1}  - a_{\,1} } \right) =   \cr 
  &  = f_{\,1} \left( {t_{\,1} } \right)
 + \delta \left( {t_{\,1}  - a_{\,1} } \right)\left( {1 - F_{\,1} \left( {t_{\,1} } \right)} \right)
 - H\left( {t_{\,1}  - a_{\,1} } \right)f_{\,1} \left( {t_{\,1} } \right) =   \cr 
  &  = \left( {1 - H\left( {t_{\,1}  - a_{\,1} } \right)} \right)f_{\,1} \left( {t_{\,1} } \right)
 + \delta \left( {t_{\,1}  - a_{\,1} } \right)\left( {1 - F_{\,1} \left( {t_{\,1} } \right)} \right) \cr} 
$$
We can check that the integral of the above properly returns the $F_m$
$$
\eqalign{
  & \int\limits_{x =  - \infty }^t {f_{\,m} \left( {x;\,a} \right)dx}  =   \cr 
  &  = \int\limits_{x =  - \infty }^t {\left[ {x < a} \right]f\left( x \right)dx}
  + \int\limits_{x =  - \infty }^t {\left( {1 - F\left( a \right)} \right)\delta \left( {x - a} \right)dx}  =   \cr 
  &  = \left[ {t < a} \right]\int\limits_{x =  - \infty }^t {f\left( x \right)dx}
  + \left[ {a \le t} \right]\int\limits_{x =  - \infty }^a {f\left( x \right)dx}
  + \left( {1 - F\left( a \right)} \right)\int\limits_{y =  - \infty }^{t - a} {\delta \left( y \right)dy}  =   \cr 
  &  = \left[ {t < a} \right]F(t) + \left[ {a \le t} \right]F(a)
 + \left( {1 - F\left( a \right)} \right)\left[ {0 \le t - a} \right] =   \cr 
  &  = \left[ {t < a} \right]F(t) + \left[ {a \le t} \right] = F_{\,m} \left( {t;\,a} \right) =   \cr 
  &  = \int\limits_{x =  - \infty }^t {\left( {1 - H\left( {x - a} \right)} \right)f\left( x \right)dx}
  + \int\limits_{x =  - \infty }^t {\left( {1 - F\left( x \right)} \right)\delta \left( {x - a} \right)dx}  =   \cr 
  &  = \left[ {t < a} \right]F(t) + \left[ {a \le t} \right]F(a)
 + \left( {1 - F\left( a \right)} \right)\left[ {0 \le t - a} \right] =   \cr 
  &  = \left[ {t < a} \right]F(t) + \left[ {a \le t} \right] = F_{\,m} \left( {t;\,a} \right) \cr} 
$$
Then the CDF of the sum of the two terms will be
$$
\eqalign{
  & P\left( {\min \left( {X_{\,1} ,a_{\,1} } \right) + \min \left( {X_{\,2} ,a_{\,2} } \right) \le s} \right)
 = F_s \left( {s;\,a_{\,1} ,a_{\,2} } \right) =   \cr 
  &  = \int\limits_t {P\left( {\min \left( {X_{\,1} ,a_{\,1} } \right) \le t} \right)
 P\left( {\min \left( {X_{\,2} ,a_{\,2} } \right) = s - t} \right)dt}  =   \cr 
  &  = \int\limits_t {\left( \matrix{
  \left( {F_{\,1} \left( t \right) + H\left( {t - a_{\,1} } \right)
 \left( {1 - F_{\,1} \left( t \right)} \right)} \right) \cdot  \hfill \cr 
  \left( \matrix{
  \left( {1 - H\left( {s - t - a_{\,2} } \right)} \right)f_{\,2} \left( {s - t} \right) +  \hfill \cr 
   + \delta \left( {s - t - a_{\,2} } \right)\left( {1 - F_{\,2} \left( {s - t} \right)} \right) \hfill \cr}  \right)
 \hfill \cr}  \right)dt}  \cr} 
$$
in the case that $X1$ and $X2$ are independent.
Otherwise we shall introduce the appropriate expressions for the CDF and PDF.
</s>