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}.

Short answer

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

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

Step-by-step solution

Content Introduction

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

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

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

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.