Questions: Expected value, variance, standard deviation

Summary

A selection of questions to test your understanding of expected values, variance, and standard deviation.

Before attempting these questions it is highly recommended that you read Guide: Expected value, variance, standard deviation.

Q1

For each of the following valid random variables with associated probability mass function, work out the expected value and variance.

1.1.

Let \(X\) be the random variable representing the result of rolling a biased four sided-die. The PMF of \(X\) is given by:

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)
\(P(X=x)\)	\(\dfrac{1}{10}\)	\(\dfrac{1}{5}\)	\(\dfrac{1}{2}\)	\(\dfrac{1}{5}\)

1.2.

A discrete random variable \(X\) has five possible outcomes (\(1, 2, 3, 4,\) or \(5\)), and the PMF is given by:

\(x\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)
\(P(X=x)\)	\(0.25\)	\(0.35\)	\(0.05\)	\(0.2\)	\(0.1\)

1.3.

A coin is tossed, where the probability of tails is \(70%\) and heads is \(30%\). Let \(X\) represent the result of the coin toss. Complete the table below:

\(x\)	Heads	Tails
\(P(X=x)\)	\(0.7\)	\(0.3\)

1.4.

The PMF for a random variable \(X\) is given as:

\(x\)	1	2	3	4
\(P(X=x)\)	\(1/10\)	\(2/10\)	\(3/10\)	\(4/10\)

Q2

For each of the following valid random variables with associated probability density function, work out the expected value and variance.

2.1.

Let \(X\) be a continuous random variable on the interval \([0, 2]\) with the PDF: \[f(x) =\begin{cases} \dfrac{1}{2} & \textsf{if } 0 \leq x \leq 2 \\[0.5em] 0 & \textsf{otherwise} \end{cases}\]

2.2.

Let \(X\) be a continuous random variable with the PDF: \[f(x) = \begin{cases} 2x & \textsf{if } 0 \leq x \leq 1, \\[0.5em] 0 & \textsf{otherwise}.\end{cases}\]

Q3

Give the expected value and variance for rolling seven fair \(6\)-sided dice. You may assume that each roll is independent of every other roll.

Q4

This question refers to the exponential distribution for a continuous random variable. You can find more information about this and Factsheet: Exponential distribution.

The PDF of the exponential distribution is \(\mathbb{P}(X=x)=\lambda e^{-\lambda x}\). Using integration by parts (see [Guide: Integration by parts]) and the fact that \[\lim_{x\to\infty}x^n e^{-\lambda x} = 0\] for any natural number \(n\) and real \(\lambda > 0\), show that

1. the mean \(\mu\) of the exponential distribution is \(\dfrac{1}{\lambda}\);

2. the variance \(\sigma^2\) of the exponential distribution is \(\dfrac{1}{\lambda^2}\).

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After attempting the questions above, please click this link to find the answers.

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Version history and licensing

v1.0: initial version created 08/25 by tdhc.

This work is licensed under CC BY-NC-SA 4.0.