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STATLECT > Additional topics in probability theory > Moment generating function

Moment generating function - Exercise set 1

This exercise set contains some solved exercises on moment generating functions. The theory needed to solve these exercises is introduced in the lecture entitled Moment generating function.

Exercise 1.1

Let X be a discrete random variable having a Bernoulli distribution. Its support R_X is:[eq1]and its probability mass function [eq2] is:[eq3]where [eq4] is a constant. Derive the moment generating function of X, if it exists.

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Using the definition of moment generating function:[eq5]Obviously, the moment generating function exists and it is well-defined because the above expected value exists for any t in R.

Exercise 1.2

Let X be a random variable with moment generating function[eq6]Derive the variance of X.

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We can use the following formula for computing the variance:[eq7]The expected value of X is computed by taking the first derivative of the moment generating function:[eq8]and evaluating it at $t=0$:[eq9]The second moment of X is computed by taking the second derivative of the moment generating function:[eq10]and evaluating it at $t=0$:[eq11]Therefore:[eq12]

Exercise 1.3

A random variable X is said to have a Chi-square distribution with n degrees of freedom if its moment generating function is defined for any $t,< rac{1}{2}$ and it is equal to:[eq13]Define [eq14]where X_1 and X_2 are two independent random variables having Chi-square distributions with $n_{1}$ and $n_{2}$ degrees of freedom respectively. Prove that Y has a Chi-square distribution with $n_{1}+n_{2}$ degrees of freedom.

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The moment generating functions of X_1 and X_2 are:[eq15]The moment generating function of a sum of independent random variables is just the product of their moment generating functions:[eq16]Therefore, [eq17] is the moment generating function of a Chi-square random variable with $n_{1}+n_{2}$ degrees of freedom. As a consequence, Y has a Chi-square distribution with $n_{1}+n_{2}$ degrees of freedom.

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