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STATLECT > Additional topics in probability theory > Legitimate probability mass functions

Legitimate probability mass functions - Exercise set 1

This exercise set contains some solved exercises on probability mass functions. The theory needed to solve these exercises is introduced in the lecture entitled Legitimate probability mass functions.

Exercise 1.1

Consider the following function:[eq1]

Prove that [eq2] is a legitimate probability mass function.

nav_button Solution

For [eq3] we have:[eq4]while for [eq5] we have:[eq6] Therefore, [eq7] for any $xin U{211d} $ and the non-negativity property is satisfied. The other necessary property (sum over the support equals 1) is also satisfied, because:[eq8]

Exercise 1.2

Consider the following function:[eq9]

Prove that [eq2] is a legitimate probability mass function.

nav_button Solution

For [eq11] we have:[eq12]while for [eq13] we have:[eq6]Therefore, [eq7] for any $xin U{211d} $ and the non-negativity property is satisfied. The other necessary property (sum over the support equals 1) is also satisfied, because:[eq16]

Exercise 1.3

Consider the following function:[eq17]

Prove that [eq2] is a legitimate probability mass function.

nav_button Solution

For $xin U{2115} $ we have:[eq19]because $4^{1-x}$ is strictly positive. For $x otin U{2115} $ we have:[eq6]Therefore, [eq7] for any $xin U{211d} $ and the non-negativity property is satisfied. The other necessary property (sum over the support equals 1) is also satisfied, because:[eq22]

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