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Normal distribution - Exercise set 1

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

Exercise 1.1

Let X be a normal random variable with mean $mu =3$ and variance $sigma ^{2}=4$. 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]

Then, we need to express the distribution function of X in terms of the distribution function of a standard normal random variable Z:[eq3]

Therefore, the above probability can be expressed as:[eq4]where we have used the fact that [eq5], which has been presented in the lecture entitled Normal distribution values.

Exercise 1.2

Let X be a random variable having a normal distribution with mean $mu =1$ and variance $sigma ^{2}=16$. Compute the following probability:[eq6]

nav_button Solution

We need to use the same technique used in the previous exercise (express the probability in terms of the distribution function of a standard normal random variable):[eq7]where we have found the value [eq8] in a normal distribution table.

Exercise 1.3

Suppose the random variable X has a normal distribution with mean $mu =1$ and variance $sigma ^{2}=1$. Define the random variable Y as follows:[eq9]Compute the expected value of Y.

nav_button Solution

Remember that the moment generating function of X is:[eq10]Therefore, using the linearity of the expected value, we obtain:[eq11]

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