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# ISI MStat PSB 2009 Problem 5 | Finding the Distribution of a Random Variable This is a very beautiful sample problem from ISI MStat PSB 2009 Problem 5 based on finding the distribution of a random variable . Let's give it a try !!

## Problem- ISI MStat PSB 2009 Problem 5

Suppose and are continuous and strictly increasing distribution
functions. Let have distribution function and (a) Find the distribution function of Y.
(b) Hence, or otherwise, show that the joint distribution function of denoted by is given by .

### Prerequisites

Cumulative Distribution Function

Inverse of a function

Minimum of two function

## Solution :

(a) Let be Cumulative distribution Function of Then , = [ taking G on both side, since G is Strictly in decreasing function the inequality doesn't change]
= [ taking on both side and since F is strictly increasing distribution function hence inverse exists and inequality doesn't change ]

= [Since F is a distribution function of X ]
=G(y)

therefore Cumulative distribution Function of is G .

(b) Let be joint cdf of then we have , = [ Since if with probability 1 then with probability 1 as F is strictly increasing distribution function ]
= = [ Since F is CDF of X ]

Therefore , the joint distribution function of denoted by is given by ## Food For Thought

Find the the distribution function of where G is continuous and strictly decreasing function .

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This is a very beautiful sample problem from ISI MStat PSB 2009 Problem 5 based on finding the distribution of a random variable . Let's give it a try !!

## Problem- ISI MStat PSB 2009 Problem 5

Suppose and are continuous and strictly increasing distribution
functions. Let have distribution function and (a) Find the distribution function of Y.
(b) Hence, or otherwise, show that the joint distribution function of denoted by is given by .

### Prerequisites

Cumulative Distribution Function

Inverse of a function

Minimum of two function

## Solution :

(a) Let be Cumulative distribution Function of Then , = [ taking G on both side, since G is Strictly in decreasing function the inequality doesn't change]
= [ taking on both side and since F is strictly increasing distribution function hence inverse exists and inequality doesn't change ]

= [Since F is a distribution function of X ]
=G(y)

therefore Cumulative distribution Function of is G .

(b) Let be joint cdf of then we have , = [ Since if with probability 1 then with probability 1 as F is strictly increasing distribution function ]
= = [ Since F is CDF of X ]

Therefore , the joint distribution function of denoted by is given by ## Food For Thought

Find the the distribution function of where G is continuous and strictly decreasing function .

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