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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 $$F$$ and $$G$$ are continuous and strictly increasing distribution
functions. Let $$X$$ have distribution function $$F$$ and $$Y=G^{-1}( F(X))$$
(a) Find the distribution function of Y.
(b) Hence, or otherwise, show that the joint distribution function of $$(X, Y),$$ denoted by $$H(x, y),$$ is given by $$H(x, y)=\min (F(x), G(y))$$.

### Prerequisites

Cumulative Distribution Function

Inverse of a function

Minimum of two function

## Solution :

(a) Let $$F_{Y}(y)$$ be Cumulative distribution Function of $$Y=G^{-1}(F(x))$$
Then , $$F_{Y}(y)=P(Y \le y) =P(G^{-1}(F(x)) \le y)$$
=$$P(F(x) \le G(y))$$

[ taking G on both side, since G is Strictly in decreasing function the inequality doesn't change]
= $$P(x \le F^{-1}(G(y)))$$

[ taking $$F^{-1}$$ on both side and since F is strictly increasing distribution function hence inverse exists and inequality doesn't change ]

=$$F(F^{-1}(G(y)))$$ [Since F is a distribution function of X ]
=G(y)

therefore Cumulative distribution Function of $$Y=G^{-1}(F(x))$$ is G .

(b) Let $$F_{H}(h)$$ be joint cdf of $$(x, y)$$ then we have ,

$$F_{H}(h)=P(X \leq x, Y \leq y) =P(X \leq x, G^{-1}(F(X)) \leq y) =P(X \leq x, F(X) \leq G(y))$$

=$$P(F(X) \leq F(x), F(X) \leq G(y))$$

[ Since if $$X \le x$$ with probability 1 then $$F(X) \le F(x)$$ with probability 1 as F is strictly increasing distribution function ]
= $$P(\min F(X) \leq \min {F(x), G(y)}) =P(X \leq F^{-1}(\min {F(x), G(y)}))$$

=$$F(F^{-1}(\min {F(x),(n(y)})) =\min {F(x), G(y)}$$ [ Since F is CDF of X ]

Therefore , the joint distribution function of $$(X, Y),$$ denoted by $$H(x, y),$$ is given by $$H(x, y)=\min (F(x), G(y))$$

## Food For Thought

Find the the distribution function of $$Y=G^{-1}( F(X))$$ where G is continuous and strictly decreasing function .

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