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# ISI MStat PSB 2008 Problem 7 | Finding the Distribution of a Random Variable

This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 7 based on finding the distribution of a random variable . Let's give it a try !!

## Problem- ISI MStat PSB 2008 Problem 7

Let $$X$$ and $$Y$$ be exponential random variables with parameters 1 and 2 respectively. Another random variable $$Z$$ is defined as follows.

A coin, with probability p of Heads (and probability 1-p of Tails) is
tossed. Define $$Z$$ by $$Z=\begin{cases} X & , \text { if the coin turns Heads } \\ Y & , \text { if the coin turns Tails } \end{cases}$$
Find $$P(1 \leq Z \leq 2)$$

### Prerequisites

Cumulative Distribution Function

Exponential Distribution

## Solution :

Let , $$F_{i}$$ be the CDF for i=X,Y, Z then we have ,

$$F_{Z}(z) = P(Z \le z) = P( Z \le z | coin turns Head )P(coin turns Head) + P( Z \le z | coin turns Tail ) P( coin turns Tail)$$

=$$P( X \le z)p + P(Y \le z ) (1-p)$$ = $$F_{X}(z)p+F_{Y}(y) (1-p)$$

Therefore pdf of Z is given by $$f_{Z}(z)= pf_{X}(z)+(1-p)f_{Y}(z)$$ , where $$f_{X} and f_{Y}$$ are pdf of X,Y respectively .

So , $$P(1 \leq Z \leq 2) = \int_{1}^{2} \{pe^{-z} + (1-p) 2e^{-2z}\} dz = p \frac{e-1}{e^2} +(1-p) \frac{e^2-1}{e^4}$$

## Food For Thought

Find the the distribution function of $$K=\frac{X}{Y}$$ and then find $$\lim_{K \to \infty} P(K >1 )$$