Q.6.10 - A First Course in Probability

Question

The joint probability density function of X and Y is given by f(x, y) = e^-(x+y)      0 … x < q, 0 … y < q Find 

(a) P{X < Y} and 

(b) P{X < a}. 

Step-by-Step Solution

Answer

a. $P\{X<Y\}=\frac{1}{2}$

b.$P\{X<a\} =1-e^{-a}$

Step 1: Content Introduction

The likelihood of an occasion is the extent of times the occasion occurs out of an enormous number of preliminaries.

Step 2: Explanation (Part a)

Observe that their joint density function can be factorized as:

$$\begin{gathered} f (x,y)=e^{-(x+y)}=e^{-x}{e}^{-y} \\ =f_X\left(x\right)f_Y\left(y\right) \end{gathered}$$

So because X and Y are equally distributed random variables with distribution Expo(1) and they are independent since the joint density function can be factorized. Because of the equal distribution, we have that

$P (X<Y)=\frac{1}{2}$

Step 3: Explanation (part b)

Use the fact that $X~Expo\left(1\right)$ to obtain that

$P(X<a)=P(X\le a)=F_X\left(a\right)=1-e^{-a}$for $a\ge 0$, otherwise it is equal to zero.