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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 !!

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 .

Cumulative Distribution Function

Inverse of a function

Minimum of two function

(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

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

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 !!

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 .

Cumulative Distribution Function

Inverse of a function

Minimum of two function

(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

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

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