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Student's t distribution - Exercise set 1

This exercise set contains some solved exercises on the Student's t distribution. The theory needed to solve these exercises is introduced in the lecture entitled Student's t distribution.

Exercise 1.1

Let X_1 be a normal random variable with mean $mu =0$ and variance $sigma ^{2}=4$. Let X_2 be a Gamma random variable with parameters $n=10$ and $h=3$, independent of X_1. Find the distribution of the ratio[eq1]

nav_button Solution

We can write:[eq2]where $Y=X_{1}/2$ has a standard normal distribution and $Z=X_{2}/3$ has a Gamma distribution with parameters $n=10$ and $h=1$. Therefore, the ratio[eq3]has a standard Student's t distribution with $n=10$ degrees of freedom and X has a Student's t distribution with mean $mu =0$, scale $sigma ^{2}=4/3$ and $n=10$ degrees of freedom.

Exercise 1.2

Let X_1 be a normal random variable with mean $mu =3$ and variance $sigma ^{2}=1$. Let X_2 be a Gamma random variable with parameters $n=15$ and $h=2$, independent of X_1. Find the distribution of the random variable[eq4]

nav_button Solution

We can write:[eq5]where $Y=X_{1}-3$ has a standard normal distribution and $Z=X_{2}/2$ has a Gamma distribution with parameters $n=15$ and $h=1$. Therefore, the ratio[eq6]has a standard Stutent's t distribution with $n=15$ degrees of freedom.

Exercise 1.3

Let X be a Student's t random variable with mean $mu =1$, scale $sigma ^{2}=4$ and $n=6$ degrees of freedom. Compute:[eq7]

nav_button Solution

First of all, we need to write the probability in terms of the distribution function of X:[eq8]Then, we express the distribution function of X in terms of the distribution function of a standard Student's t random variable Z with $n=6$ degrees of freedom:[eq9]so that:[eq10]where the difference [eq11] can be computed with a computer algorithm, for example using the MATLAB command[eq12]

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