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STATLECT > Fundamentals of probability > Expected value

Expected value - Exercise set 2

This exercise set contains some solved exercises on the expected value of absolutely continuous random variables. The theory needed to solve these exercises is introduced in the lecture entitled Expected value.

Exercise 2.1

Let X be an absolutely continuous random variable with uniform distribution on the interval $left[ 1,3ight] $.

Its support is:[eq1]

Its probability density function is:[eq2]

Compute the expected value of X.

nav_button Solution

Since X is absolutely continuous, its expected value can be computed as an integral:[eq3]

Note that the trick is to: 1) subdivide the interval of integration to isolate the sub-intervals where the density is zero; 2) split up the integral among the various sub-intervals.

Exercise 2.2

Let X be an absolutely continuous random variable. Its support is:[eq4]

Its probability density function is:[eq5]

Compute the expected value of X.

nav_button Solution

Since X is absolutely continuous, its expected value can be computed as an integral:[eq6]

Exercise 2.3

Let X be an absolutely continuous random variable. Its support is:[eq7]

Its probability density function is:[eq8]

Compute the expected value of X.

nav_button Solution

Since X is absolutely continuous, its expected value can be computed as an integral:[eq9]

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