Basic probabilistic inequalities - Exercise set 1

This exercise set contains some solved exercises on probabilistic inequalities (Markov's inequality, Chebyshev's inequality, Jensen's inequality). The theory needed to solve these exercises is introduced in the lecture entitled Basic probabilistic inequalities.

Exercise 1.1

Let be a positive random variable whose expected value is:Find a lower bound to the following probability:

Solution

First of all, we need to use the formula for the probability of a complement:Now, we can use Markov's inequality: [eq4] Multiplying both sides of the inequality by , we obtainAdding to both sides of the inequality we obtain

Thus, the lower bound is

Exercise 1.2

Let be a random variable such that [eq8] Find a lower bound to its variance.

Solution

The lower bound can be derived thanks to Chebyshev's inequality: [eq9] Thus, the lower bound is:

Exercise 1.3

Let be a strictly positive random variable, such that [eq11] What can you infer, using Jensen's inequality, about the following expected value:

Solution

The functionhas first derivativeand second derivative [eq15] The second derivative is strictly negative on the domain of definition of the function. Therefore, the function is strictly concave. Furthermore, is not almost surely constant, because it has strictly positive variance. Hence, by Jensen's inequality:

by Marco Taboga

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