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# ISI MStat PSB 2013 Problem 8 | Finding the Distribution of a Random Variable This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 8 based on finding the distribution of a random variable . Let's give it a try !!

## Problem- ISI MStat PSB 2013 Problem 8

1. Suppose is a standard normal random variable. Define
2. (a) Show that is also a standard normal random variable.
(b) Obtain the cumulative distribution function of in terms of the cumulative distribution function of a standard normal random
variable.

### Prerequisites

Cumulative Distribution Function

Normal Distribution

## Solution :

(a) Let be distribution function of X_{2}\) then we can say that , = = = Since hence it's symmetric about 0 . So, and are identically distributed .

Therefore , = Hence , is also a standard normal random variable.

(b) Let , Distribution function = = \)

= = = = .

## Food For Thought

Find the the distribution function of in terms of the cumulative distribution function of a standard normal random variable.

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This is a very beautiful sample problem from ISI MStat PSB 2013 Problem 8 based on finding the distribution of a random variable . Let's give it a try !!

## Problem- ISI MStat PSB 2013 Problem 8

1. Suppose is a standard normal random variable. Define
2. (a) Show that is also a standard normal random variable.
(b) Obtain the cumulative distribution function of in terms of the cumulative distribution function of a standard normal random
variable.

### Prerequisites

Cumulative Distribution Function

Normal Distribution

## Solution :

(a) Let be distribution function of X_{2}\) then we can say that , = = = Since hence it's symmetric about 0 . So, and are identically distributed .

Therefore , = Hence , is also a standard normal random variable.

(b) Let , Distribution function = = \)

= = = = .

## Food For Thought

Find the the distribution function of in terms of the cumulative distribution function of a standard normal random variable.

## Subscribe to Cheenta at Youtube

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