Q.6.6 - A First Course in Probability

Question

If X and Y are jointly continuous with joint density function fX,Y(x, y), show that X + Y is continuous with density function $fX+Y^{\left(t\right)} = q -q fX,Y(x, t - x) dx$

Step-by-Step Solution

Answer

Find the CDF of X + Y first, and then use the theorem about the derivation of function where the argument is in the boundary to obtain the required PDF.

Step 1: Content Introduction

The derivative of the CDF is the probability density function f(x), abbreviated PDF if it exists. A distribution function FX describes each random variable X. (x).

Step 2: Content Explanation

Lets find the CDF of Z:= X + Y firstly. Take any z. We have that

$$\begin{gathered} F_z\left(z\right)=P (Z\le z) \\ =P(X+Y\le z)={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{x-z}{f}_{X, Y}(x,y)dxdy \end{gathered}$$

Using the theorem from analysis about the derivation of function where the argument is in the boundary of integral, we have that

$f_z\left(z\right)=\frac{d}{dz}{F}_z\left(z\right)={\int }_{-\infty }^{\infty }{F}_{X,Y}(x,x-z)dx$

which had to be proved.