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

Legitimate probability density functions - Exercise set 1

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

Exercise 1.1

Consider the following function:[eq1]

where [eq2]. Prove that [eq3] is a legitimate probability density function.

nav_button Solution

Since $lambda >0$ and the exponential function is strictly positive, [eq4] for any $xin U{211d} $, so the non-negativity property is satisfied. The integral property is also satisfied, because:[eq5]

Exercise 1.2

Consider the following function:[eq6]

where $l,uin U{211d} $ and $l<u$. Prove that [eq7] is a legitimate probability density function.

nav_button Solution

$l<u$ implies $ rac{1}{u-l}>0$, so [eq8] for any $xin U{211d} $ and the non-negativity property is satisfied. The integral property is also satisfied, because:[eq9]

Exercise 1.3

Consider the following function:[eq10]where $nin U{2115} $ and [eq11] is the Gamma function. Prove that [eq12] is a legitimate probability density function.

nav_button Solution

Remember the definition of Gamma function:[eq13][eq14] is obviously strictly positive for any $z$, since [eq15] is strictly positive and $x^{z-1}$ is strictly positive on the interval of integration (except at 0 where it is 0). Therefore, [eq12] satisfies the non-negativity property, because the four factors in the product[eq17]are all non-negative on the interval [eq18].

The integral property is also satisfied, because:[eq19]

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