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Chi-square distribution - Exercise set 1

This exercise set contains some solved exercises on the Chi-square distribution. The theory needed to solve these exercises is introduced in the lectures entitled Chi-square distribution and Chi square distribution values.

Exercise 1.1

Let X be a chi-square random variable with $3$ degrees of freedom. Compute the following probability:[eq1]

nav_button Solution

First of all, we need to express the above probability in terms of the distribution function of X:[eq2]where the values[eq3]can be computed with a computer algorithm or found in a Chi-square distribution table (see the lecture entitled Chi-square distribution values).

Exercise 1.2

Let X_1 and X_2 be two independent normal random variables having mean $mu =0$ and variance $sigma ^{2}=16$. Compute the following probability:[eq4]

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First of all, the two variables X_1 and X_2 can be written as:[eq5]where $Z_{1}$ and $Z_{2}$ are two standard normal random variables. Thus, we can write:[eq6]but the sum [eq7] has a Chi-square distribution with $2$ degrees of freedom. Therefore:[eq8]where [eq9] is the distribution function of a Chi-square random variable Y with $2$ degrees of freedom, evaluated at the point $y= rac{1}{2}$. Using any computer program, we can find:[eq10]

Exercise 1.3

Suppose the random variable X has a Chi-square distribution with $5$ degrees of freedom. Define the random variable Y as follows:[eq11]Compute the expected value of Y.

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The expected value of Y can be easily calculated using the moment generating function of X:[eq12]Now, exploiting the linearity of the expected value, we obtain:[eq13]

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