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STATLECT > Additional topics in probability theory > Functions of random variables and their distribution

Functions of random variables and their distribution - Exercise set 1

This exercise set contains some solved exercises on how to derive the distribution of a function of a random variable. The theory needed to solve these exercises is introduced in the lecture entitled Functions of random variables and their distribution.

Exercise 1.1

Let X be an absolutely continuous random variable with support[eq1]and probability density function:[eq2]Let [eq3]Find the probability density function of Y.

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The support of Y is:[eq4]The function $g$ is strictly increasing and its inverse is[eq5]with derivative[eq6]The probability density function of Y is:[eq7]

Exercise 1.2

Let X be an absolutely continuous random variable with support[eq8]and probability density function:[eq9]Let [eq10]Find the probability density function of Y.

nav_button Solution

The support of Y is:[eq11]The function $g$ is strictly decreasing and its inverse is[eq12]with derivative[eq13]The probability density function of Y is:[eq14]

Exercise 1.3

Let X be a discrete random variable with support[eq15]and probability mass function:[eq16]Let [eq17]Find the probability mass function of Y.

nav_button Solution

The support of Y is:[eq18]The function $g$ is strictly increasing and its inverse is[eq19]The probability mass function of Y is:[eq20]

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