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Gamma distribution - Exercise set 1

This exercise set contains some solved exercises on the Gamma distribution. The theory needed to solve these exercises is introduced in the lecture entitled Gamma distribution.

Exercise 1.1

Let X_1 and X_2 be two independent Chi-square random variables having $3$ and $5$ degrees of freedom respectively. Consider the following random variables:[eq1]What distribution do they have?

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Being multiples of Chi-square random variables, the variables $Y_{1}$, $Y_{2}$ and $Y_{3}$ all have a Gamma distribution. The random variable X_1 has $n=3$ degrees of freedom and the random variable $Y_{1}$ can be written as[eq2]where $h=6$. Therefore $Y_{1}$ has a Gamma distribution with parameters $n=3$ and $h=6$. The random variable X_2 has $n=5$ degrees of freedom and the random variable $Y_{2}$ can be written as[eq3]where $h=5/3$. Therefore $Y_{2}$ has a Gamma distribution with parameters $n=5$ and $h=5/3$. The random variable $X_{1}+X_{2}$ has a Chi-square distribution with $n=3+5=8$ degrees of freedom, because X_1 and X_2 are independent (see the lecture entitled Chi-square distribution), and the random variable $Y_{3}$ can be written as[eq4]where $h=24$. Therefore $Y_{3}$ has a Gamma distribution with parameters $n=8$ and $h=24$.

Exercise 1.2

Let X be a random variable having a Gamma distribution with parameters $n=4 $ and $h=2$. Define the following random variables:[eq5]

What distribution do these variables have?

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Multiplying a Gamma random variable by a strictly positive constant one still obtains a Gamma random variable. In particular, the random variable $Y_{1}$ is a Gamma random variable with parameters $n=4$ and [eq6] The random variable $Y_{2}$ is a Gamma random variable with parameters $n=4$ and [eq7] The random variable $Y_{3}$ is a Gamma random variable with parameters $n=4$ and [eq8]The random variable $Y_{3}$ is also a Chi-square random variable with $4$ degrees of freedom (remember that a Gamma random variable with parameters n and $h$ is also a Chi-square random variable when $n=h$).

Exercise 1.3

Let X_1, X_2 and $X_{3}$ be mutually independent normal random variables having mean $mu =0$ and variance $sigma ^{2}=3$. Consider the random variable[eq9]What distribution does X have?

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The random variable X can be written as [eq10]where $Z_{1}$, $Z_{2}$ and $Z_{3}$ are mutually independent standard normal random variables. The sum [eq11] has a Chi-square distribution with $3$ degrees of freedom (see the lecture entitled Chi-square distribution). Therefore X has a Gamma distribution with parameters $n=3$ and $h=18$.

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