2.3: Expected Value and Variance

Additional Reference: See Introduction to Random Variables where we discovered some important properties for a random variable $X$ that are summarized in the Appendix.

How Much is the Raffle Ticket Worth?

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Image credit: Randy Heinitz via flickr

A raffle sells 1,000 tickets with the following payouts:

• There is one winning randomly selected ticket that will win the grand prize of $5,000.
• Two tickets will be randomly selected to win a second prize each worth $1,000.
• Ten tickets will be randomly selected to win a third prize each worth $200.
• The remaining tickets do not win a prize.

Question 1

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Let random variable $X$ be the amount of money won by a randomly selected raffle ticket. We let $p(x)$ denote the corresponding probability mass function of $X$.

Question 1a

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Fill in the values of $x$ and $p(x)$ in the table below.

Solution to Question 1a

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Fill in the blanks to complete the table.

$x$	??	??	??	??
$p(x)$	??	??	??	??

Question 1b

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If somebody offered to buy a raffle ticket from you, what do you think fair value is for the ticket?

Solution to Question 1b

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Question 1c

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Consider another ticket raffle. There are 1,000 tickets with the following payouts:

• 500 randomly selected tickets will win a grand prize of $15.
• The other 500 tickets do not win a prize.

Let random variable $Y$ be the amount of money won by a randomly selected raffle ticket.

Somebody offers one free ticket to either raffle $X$ or raffle $Y$ (but not both!), which would you prefer and why?

Solution to Question 1c

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The Expected Value of a Discrete Random Variable

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The expected value for a discrete random variable $X$ is denoted $E(X)$. We compute

$$E(X)={\mu }_X=\sum _x(x\cdot p(x)).$$

• $E(X)$ is the average or mean value of random variable $X$.
• The expected value of $X$ is denoted $E(X)$ or ${\mu }_X$.
• The expected value of $X$ might not be a possible value of $X$.

The Variance of a Discrete Random Variable

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The variance for a discrete random variable $X$ is one common way to measure how spread out (in relation to the expected value) are the values of $X$. We compute

$$\text{Var}(X)={\sigma }_X^2=E((x-{\mu }_X{)}^2)=\sum _x((x-{\mu }_X{)}^2\cdot p(x)).$$

• $\text{Var}(X)$ is the expected value of the squared distance from the mean.
• The variance is denoted $\text{Var}(X)$ or ${\sigma }_X^2$.

The Standard Deviation of a Discrete Random Variable

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The standard deviation for a discrete random variable $X$ is the square root of the variance,

$$\text{SD}(X)={\sigma }_X=\sqrt{\text{Var}(X)}.$$

• The standard deviation of random variable $X$ is denoted ${\sigma }_X$.
• The standard deviation essentially measures the average distance of the values of $X$ from its mean ${\mu }_X$.
• The units of ${\sigma }_X$ and $X$ are the same, and thus standard deviation is often a practical way to describe the spread of $X$.

Question 2

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Using properties of discrete random variables, show that for any discrete random variable $X$ with pmf $p(X)$ and expected value $E(X)=\mu$, we have

$$\text{Var}(X)=E(X^2)-{\mu }^2.$$

Solution to Question 2

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Finish the proof below!

Let $X$ be a discrete random variable with pmf $p(x)$ and expected value $E(X)=\mu$. Then we have

$$\begin{array}{rcll}\text{Var}(X)& =& \sum _x((x-\mu {)}^2\cdot p(x))& \text{(by definition)}\\ & =& \sum _x((x^2-2\mu x+{\mu }^2)\cdot p(x))& \text{(distributing squared term)}\\ & =& \sum _x(x^2\cdot p(x)-2\mu x\cdot p(x)+{\mu }^2\cdot p(x))& \text{(distributing p(x))}\\ & =& \sum _x(x^2\cdot p(x))-\sum _x(2\mu x\cdot p(x))+\sum _x({\mu }^2\cdot p(x))& \text{(properties of summation)}\\ & =& ??& \text{(??)}\\ & =& ??& \text{(??)}\\ & =& ??& \text{(??)}\\ & =& ??& \text{(??)}\end{array}$$

Therefore, we see that $\text{Var}(X)=E(X^2)-{\mu }^2$.

Question 3

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Let $X$ and $Y$ denote the raffle ticket random variables from Question 1.

Question 3a

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Calculate $E(X)$ and $\text{Var}(X)$.

Solution to Question 3a

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Question 3b

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Calculate $E(Y)$ and $\text{Var}(Y)$.

Solution to Question 3b

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Mean, Median, and Variance of Continuous Random Variables

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Let $f(x)$ and $F(x)$ denote the probability density function and cumulative distribution function for a continuous random variable $X$.

• The expected value or mean is

$$E(X)={\mu }_X={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}x\cdot f(x)dx.$$

• The variance is

$$\text{Var}(X)=E[(X-{\mu }_X{)}^2]=E(X^2)-{\mu }_X^2.$$

• The median is the value $x$ such that $P(X<x)=0.5$. Thus, to find the median we solve the equations below for $x$,

$${\int }_{-\mathrm{\infty }}^{x}f(t)dt=0.5\text{or equivalently}F(x)=0.5.$$

Question 4

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Consider the random variable $X$ with pdf

$$f_X(x)=\{\begin{array}{ll}\frac{x}{8},& 0\le x\le 4\\ 0,& \text{otherwise}\end{array}.$$

Question 4a

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On a separate piece of paper, sketch a graph of the pdf, $f_X$.

Solution to Question 4a

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Sketch a graph on a separate piece of paper.

Question 4b

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Enter the formula for $F_X$ below. Then on a separate piece of paper sketch the graph $F_X$.

Solution to Question 4b

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$$F_X(x)=\{\begin{array}{ll}0& x<0\\ ??& 0\le x\le 4\\ 1& x>4\end{array}$$

Sketch a graph on a separate piece of paper.

Question 4c

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Calculate $P(X<1)$ and illustrate this value on each of your graphs in the solutions to Question 4a and Question 4b.

Solution to Question 4c

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Question 4d

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Calculate $E(X)$ and illustrate this on your graph in the solution to Question 4a.

Solution to Question 4d

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Question 4e

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Give the median value and illustrate this value on both of your graphs in the solutions to Question 4a and Question 4b.

Solution to Question 4e

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Question 4f

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Compute $\text{Var}(X)$.

Solution to Question 4f

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Question 5

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Let $X$ and $Y$ denote the raffle ticket random variables from Question 1.

Question 5a

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Do you believe random variables $X$ and $Y$ are independent random variables? Explain why or why not.

Solution to Question 5a

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Question 5b

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If you purchase 3 raffle tickets from raffle $X$ and 2 raffle tickets from raffle $Y$, what is the expected value of your winnings?

Solution to Question 5b

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Question 5c

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If you purchase 3 raffle tickets from raffle $X$ and 2 raffle tickets from raffle $Y$, what is the variance of your winnings?

Solution to Question 5c

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Properties of Expected Value

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Let $X$ and $Y$ denote a random variables, and let $a$ and $b$ denote constants. Then we have the following properties.

• $E(a)=a$
• $E(aX+bY)=aE(X)+bE(Y)$

Properties of Variance

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• For any random variable $X$, we have $\text{Var}(X)=E(X^2)-{\mu }_X^2$.

• If $X$ and $Y$ are independent random variables and $a$ and $b$ constants, then $\text{Var}(aX+bY)=a^2\text{Var}(X)+b^2\text{Var}(Y)$.

Question 6

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The data set spotify-hits.csv¹ is stored online and contains audio statistics of the top 2000 tracks on Spotify from 2000-2019. The data is stored in a comma separated file (csv).

• We can use the function read.csv() to import the csv file into an R data frame we call hits.

hits <- read.csv("https://raw.githubusercontent.com/CU-Denver-MathStats-OER/Statistical-Theory/main/Data/spotify-hits.csv")

In the code cell below:

• We convert artist, song, and genre to categorical variables using the factor() function.
• Extract the variables artist, song, energy, acousticness, and genre (ignoring the rest).
• Print the first 6 rows to screen to get a glimpse of the resulting data frame.

hits$artist <- factor(hits$artist)  # artist is categorical
hits$song <- factor(hits$song)  # song is categorical
hits$genre <- factor(hits$genre)  # genre is categorical
hits <- hits[,c("artist", "song", "energy", "acousticness", "genre")] 
head(hits)  # display first 6 rows of data frame

          artist                   song energy acousticness             genre
1 Britney Spears Oops!...I Did It Again  0.834       0.3000               pop
2      blink-182   All The Small Things  0.897       0.0103         rock, pop
3     Faith Hill                Breathe  0.496       0.1730      pop, country
4       Bon Jovi           It's My Life  0.913       0.0263       rock, metal
5         *NSYNC            Bye Bye Bye  0.928       0.0408               pop
6          Sisqo             Thong Song  0.888       0.1190 hip hop, pop, R&B

• Energy: A measure of how energetic a song is from $0.0$ to $1.0$ (least to most energy) of. Typically, energetic songs are fast, loud, and noisy.
• Acousticness: A measure from $0.0$ to $1.0$ (least to most acoustic) of depending on how significant the use of acoustic instruments are in the song.

Let $X$ denote the energy of a randomly selected song, and let $Y$ denote the acousticness of a randomly selected song. We define a new song metric, $Z$, that is a weighted mean of score $X$ and $Y$:

$$Z=\frac{3X+2Y}{5}$$

Question 6a

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Do you believe $X$ and $Y$ are independent random variables? Explain why or why not.

Solution to Question 6a

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Question 6b

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Use R to compute $E(X)$, $E(Y)$, and $E(Z)$. Check whether or not the property $E(aX+bY)=aE(X)+bE(Y)$ holds in this context.

• Hint: Recall R, the function mean(x) calculates the mean (expected value) of x.

Solution to Question 6b

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x <- hits$energy  # random variable x
y <- hits$acousticness  # random variable y
z <- (3*x + 2*y) / 5  # random variable z

# use code cell to compare expected values

Question 6c

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Use R to compute $\text{Var}(X)$, $\text{Var}(Y)$, and $\text{Var}(Z)$. Check whether or not the property $\text{Var}(aX+bY)=a^2\text{Var}(X)+b^2\text{Var}(Y)$ holds in this context.

• Hint: The function var(x) calculates the variance of x.

Solution to Question 6c

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# use code cell to compare variances

Question 6d

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Determine whether each of the statements below are true or false. If false, explain why.

For any two random variables $X$ and $Y$ and constants $a$ and $b$:

• It always follows that $E(aX+bY)=aE(X)+bE(Y)$.
• It always follows that $\text{Var}(aX+bY)=a^2\text{Var}(X)+b^2\text{Var}(Y)$.

Question 6d

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Appendix: Properties of Random Variables

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In Introduction to Random Variables we discovered the following properties for a random variable $X$.

Properties of Discrete Random Variables

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For a discrete random variable $X$, let $p(x)$ and $F(x)$ denote the pmf and cdf, respectively, we have:

• $0\le p(x)\le 1$ for all $x$

• ${\displaystyle \sum _{\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{x}}p(x)=1}$

• $F(x)={\displaystyle P(X\le x)=\sum _{k=x_{\mathrm{m}\mathrm{i}\mathrm{n}}}^{x}p(k)}$

• $0\le F(x)\le 1$ for all $x$

• ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}F(x)=1}$

• $F(x)$ is nondecreasing.

Properties of Continuous Random Variables

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For a continuous random variable $X$, let $f(x)$ and $F(x)$ denote the pdf and cdf, respectively, we have:

• $f(x)\ge 0$ for all $x$

• ${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)=1}$

• ${\displaystyle P(a<x<b)={\int }_a^{b}f(x)dx}$

• ${\displaystyle F(x)={\int }_{-\mathrm{\infty }}^{x}f(t)dt}$

• The $F(x)$ is an antiderivative of $f$.

• The $f(x)$ is the derivative of $F(x)$.

• $0\le F(x)\le 1$ for all $x$.

• ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}F(x)=1}$.

• $F(x)$ is nondecreasing.

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Statistical Methods: Exploring the Uncertain by Adam Spiegler is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

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