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# ISI MStat PSB 2004 Problem 7 | Finding the Distribution of a Random Variable This is a very beautiful sample problem from ISI MStat PSB 2004 Problem 7 based on finding the distribution of a random variable . Let's give it a try !!

## Problem- ISI MStat PSB 2004 Problem 7

Suppose X has a normal distribution with mean 0 and variance 25 . Let Y be an independent random variable taking values -1 and 1 with equal probability. Define S=XY+XY$S=X Y+\frac{X}{Y}$ and T=XYXY$T=X Y-\frac{X}{Y}$
(a) Find the probability distribution of s.
(b) Find the probability distribution of (S+T10)2$(\frac{S+T}{10})^{2}$

### Prerequisites

Cumulative Distribution Function

Normal distribution

## Solution :

(a) We can write Let Cumulative distribution function of S be denoted by . Then , ----(1)

Here given that Y takes values 1 and -1 with equal probabilities .so , .

Now as hence X is symmetric distribution about 0 . Thus X and -X are identically distributed .

Thus from (1) we get = Hence .

(b) Let K= = Let C.D.F of K be = as .

## Food For Thought

Find the distribution of T .

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This is a very beautiful sample problem from ISI MStat PSB 2004 Problem 7 based on finding the distribution of a random variable . Let's give it a try !!

## Problem- ISI MStat PSB 2004 Problem 7

Suppose X has a normal distribution with mean 0 and variance 25 . Let Y be an independent random variable taking values -1 and 1 with equal probability. Define S=XY+XY$S=X Y+\frac{X}{Y}$ and T=XYXY$T=X Y-\frac{X}{Y}$
(a) Find the probability distribution of s.
(b) Find the probability distribution of (S+T10)2$(\frac{S+T}{10})^{2}$

### Prerequisites

Cumulative Distribution Function

Normal distribution

## Solution :

(a) We can write Let Cumulative distribution function of S be denoted by . Then , ----(1)

Here given that Y takes values 1 and -1 with equal probabilities .so , .

Now as hence X is symmetric distribution about 0 . Thus X and -X are identically distributed .

Thus from (1) we get = Hence .

(b) Let K= = Let C.D.F of K be = as .

## Food For Thought

Find the distribution of T .

## Subscribe to Cheenta at Youtube

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