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# ISI MStat PSB 2012 Problem 3 | Finding the Distribution of a Random Variable This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 3 based on finding the distribution of a random variable . Let's give it a try !!

## Problem- ISI MStat PSB 2012 Problem 3

Let and be i.i.d. exponential random variables with mean .Let and where is a Bernoulli random variable with parameter and is independent of and (a) Show that and have the same distribution.
(b) Obtain the common density function.

### Prerequisites

Cumulative Distribution Function

Bernoulli distribution

Exponential Distribution

## Solution :

Cumulative distribution of be ,  Now, Now, if then, = = = = Now, then,     Therefore, Cumulative distribution of be , =  = = since cdf of exponential random Variable, X is Thus both and has same distribution
(b) = Similarly, for .

## Food For Thought

If then find the distribution of , where .

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This is a very beautiful sample problem from ISI MStat PSB 2012 Problem 3 based on finding the distribution of a random variable . Let's give it a try !!

## Problem- ISI MStat PSB 2012 Problem 3

Let and be i.i.d. exponential random variables with mean .Let and where is a Bernoulli random variable with parameter and is independent of and (a) Show that and have the same distribution.
(b) Obtain the common density function.

### Prerequisites

Cumulative Distribution Function

Bernoulli distribution

Exponential Distribution

## Solution :

Cumulative distribution of be ,  Now, Now, if then, = = = = Now, then,     Therefore, Cumulative distribution of be , =  = = since cdf of exponential random Variable, X is Thus both and has same distribution
(b) = Similarly, for .

## Food For Thought

If then find the distribution of , where .

## Subscribe to Cheenta at Youtube

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