Wize University Statistics Textbook > Discrete Random Variables

Randomness and Probability

Discrete Random Variables

A random variable $X$ takes on values that depend on the outcomes of a random experiment.

Each value this variable takes on depends on a probability (outcomes cannot be determined with certainty).
The values of a random variable can vary with each repetition of the experiment.

A discrete random variable can only take on a finite number of values (it has a countable number of possible values).

Example

Which of the following is/are discrete random variables?

1. $X$ represents the number of heads that shows up in 3 consecutive tosses of a fair coin. → ________

Tossing a fair coin 3 times is a random experiment, each toss of a head has a probability of $\frac{1}{2}$
$X$ can take on four possible values (0 heads, 1 head, 2 heads, or 3 heads), each with a specific probability

2. $X$ represents the number of correct answers a student gets by randomly guessing the answers to 10 multiple choice questions, each with 1 correct option and 3 incorrect ones. → ________

Guessing the answer 10 times is a random experiment, each correct answer has a probability of $\frac{1}{4}$
$X$ can take on 11 possible values (0 correct, 1 correct, 2 correct, ... , 10 correct), each with a specific probability

3. $X$ represents the height of a customer who is randomly selected at a grocery store. → ________

Although randomly selecting a customer and measuring their height is a random experiment, $X$ does NOT take on a finite (countable) number of possible values since a customer can take on any height within a certain range (ex. they can be 151cm, 151.2cm, 151.22cm, 151.222cm, etc.)

Practice: Discrete Random Variables

Which of the following are discrete random variables? (Check all that applies.)

The number of times a cat uses the litter box in a day.

The time it takes to drive to work.

The number of spoiled blueberries in a box.

Starting salary of recent university graduates.

The number of earthquakes that occur in Alaska.

I don't know

Probability Distribution Function

The probability distribution function (PDF) of a discrete random variable $X$ lists all possible values this variable can take, along with their associate probabilities.

The PDF of a discrete random variable is usually expressed as a table:

\begin{array}{|c|c|} \hline x & P(X=x)\\ \hline \\ x_1&P(X=x_1)\\ \\ x_2&P(X=x_2)\\ \\ x_3&P(X=x_3)\\ \\ \vdots&\vdots \\ \\ \begin{array}{cc}\text{all possible values}\\ \text{that X can take on}\end{array} & \begin{array}{cc}\text{the probability that}\\\text{ X takes on the value }\\ \text{in the }x\text{ column}\end{array}\\ \\ \hline \end{array}

Notes

$0\leq P(x_i)\leq{1\rightarrow}$ each probability is between 0 and 1
$\displaystyle \sum P(x_i)={1\rightarrow}$ the sum of all probabilities is equal to 1

Example

A bird laid 4 eggs. Let $X$ be the number of baby birds that are born (hatched) healthy.

Here's the PDF of $X$ :

$\begin{array}{|c|c|} \hline x&P(X=x)\\ \hline 0&0.1\\ \hline 1&0.2\\ \hline 2&0.4\\ \hline 3&0.2\\ \hline 4&0.1\\ \hline \end{array}$

$X$ can only take on a finite number of possible values (the bird can have any whole number of healthy babies between 0 and 4)
All the possible values are listed
$x=\left\{0,1,2,3,4\right\}$
The probabilities associated with each possible $x$ is listed and $0\le P(x_i)\le1$
The sum of all probabilities $P(x_i)$ 's is equal to 1

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Expected Value, Variance, and Standard Deviation of Discrete Random Variables

Let $X$ be a discrete random variable.

The expected value or mean of $X$ is denoted by $E(X)$ or $\mu$ , and is calculated by

\boxed{E(X)=\mu=\sum \left[x_i\times P(x_i) \right]}

The variance of $X$ is denoted by $Var(X)$ or $\sigma^2$ , and is calculated by

\boxed{ \begin{array}{c} Var(X)=\sigma^2=\sum\left[(x_i-\mu)^2\times P(x_i)\right]\\ \\ \text{or}\\ \\ Var(X)=\sigma^2=E(X^2)-\mu^2\\ \\ \text{or}\\ \\ Var(X)=\sigma^2=\left[\sum x_i^{\ 2}\times P(x_i)\right]-\mu^2 \end{array}}

The standard deviation of $X$ is denoted by $SD(X)$ or $\sigma$ , and is calculated by

\boxed{SD(X)=\sigma=\sqrt{Var(X)}}

Example

Let $X$ be the random variable that represents the number of Heads that comes up in 2 consecutive tosses of a fair coin. Find the mean, variance, and standard deviation of $X$ .

Here's the PDF of X, along with some additional columns that will help with the calculation of mean, variance, and standard deviation.

TT: $x=0$ Heads; $P(X=0)=\frac{1}{4}$
TH or HT: $x=1$ Heads; $P\left(X=1\right)=\frac{1}{4}+\frac{1}{4}=\frac{2}{4}$
HH: $x=2$ Heads; $P\left(X=2\right)=\frac{1}{4}$

$\begin{array}{c|c|c|c|c} x&P[X=x]&\color{orange}x_i\cdot P(x_i)&\color{green}x^2 &\color{blue}x_i^2\cdot P(x_i)\\ \hline\\ 0&1/4&\color{orange}0&\color{green}0^2=0&\color{blue}0\times1/4=0\\ \\ 1&2/4&\color{orange}2/4&\color{green}1^2=1&\color{blue}1\times2/4=2/4\\ \\ 2&1/4&\color{orange}2/4&\color{green}2^2=4&\color{blue}4\times1/4=4/4 \end{array}$

Mean

$E(X)=\mu=\sum_{ }^{ }\left[x_i\cdot P(x_i)\right]$

$\displaystyle \mu=0+\frac{2}{4}+\frac{2}{4}$

$\mu=1$

Variance

$Var(X)=\left[\sum_{ }^{ }x_i^{\ 2}\cdot P(x_i)\right]-\mu^2$

$\displaystyle Var(X)=\left[0+\frac{2}{4}+\frac{4}{4}\right]-\left(1\right)^2$

$\displaystyle Var(X)=\frac{6}{4}-1$

$\displaystyle Var(X)=\frac{1}{2}$

Standard Deviation

$SD(X)\text{ or }\sigma=\sqrt{Var\left(X\right)}$

$=\sqrt{\frac{1}{2}}$

Example: Expected Value & Standard Deviation

Suppose you roll a fair die once. Let X be the number you get.

Find the expected value, variance, and standard deviation of $X$ .

All the possible outcomes: $x=\left\{1,2,3,4,5,6\right\}$

Here is the PDF:

Expected Value or Mean

$\displaystyle E\left(X\right)=\sum_{ }^{ }\left(x_i\right)P\left(X=x_i\right)$

$\displaystyle E\left(X\right)=\left(1\right)\left(\frac{1}{6}\right)+\left(2\right)\left(\frac{1}{6}\right)+\left(3\right)\left(\frac{1}{6}\right)+\left(4\right)\left(\frac{1}{6}\right)+\left(5\right)\left(\frac{1}{6}\right)+\left(6\right)\left(\frac{1}{6}\right)$

$E(X)=3.5$

Variance

Method 1:

$\displaystyle \sigma_x^2$

$\displaystyle=\left(\sum_{ }^{ }\left(x_i^2\right)P\left(X=x_i\right)\right)-\left(\mu_x\right)^2\ \$

$\displaystyle =\left[\left(1\right)^2\left(\frac{1}{6}\right)+\left(2\right)^2\left(\frac{1}{6}\right)+\left(3\right)^2\left(\frac{1}{6}\right)+\left(4\right)^2\left(\frac{1}{6}\right)+\left(5\right)^2\left(\frac{1}{6}\right)+\left(6\right)^2\left(\frac{1}{6}\right)\right]-\left(3.5\right)^2$

$=2.9167$

Method 2:

$\displaystyle \sigma_x^2$

$\displaystyle=\sum_{ }^{ }\left(x_i-\mu_x\right)^2P\left(X=x_i\right)$

$\displaystyle=\left(1-3.5\right)^2\left(\frac{1}{6}\right)+\left(2-3.5\right)^2\left(\frac{1}{6}\right)+\left(3-3.5\right)^2\left(\frac{1}{6}\right)+\left(4-3.5\right)^2\left(\frac{1}{6}\right)+\left(5-3.5\right)^2\left(\frac{1}{6}\right)+\left(6-3.5\right)^2\left(\frac{1}{6}\right)$

$=2.9167$

Standard Deviation

$\sigma_x=\sqrt{2.9167}=1.7078$

Example: Probability Distribution Function

The exact number of cookies in a box varies from box to box according to the probability distribution function (PDF) given by:

(a) What is the probability of getting exactly 20 cookies?

Find the probability of 20 using the complement rule:

P\left(x=20\right)=1-all\ other\ probabilities=1-0.30-0.22-0.10-0.18-0.08=0.12

(b) Find the probability that there are exactly 18 cookies given that there are at least 18 cookies in a jar.

Conditional probability:

P\left(18\ cookies|\ at\ least\ 18\ cookies\right)=\frac{P\left(X=18\right)}{P\left(X\geqslant18\right)}=\frac{0.18}{0.18+0.08+0.12}=0.474

(c) If in fact there are at most 17 cookies in a jar, what is the probability that there will be at least 16 cookies in the jar?

Conditional probability:

P(at\ least\ 16\ |\ at\ most\ 17)=\frac{P\left(X\geqslant16\right)}{P\left(X\leqslant17\right)}=\frac{0.22+0.10}{0.30+0.22+0.10}=0.516

Amori is good at math and spelling so she entered a math contest and a spelling bee. The probability of her winning each contest is listed below. Assume the two contests are independent.

Illustration for practice question: Discrete Random Variable

(a) What is the probability that she wins both contests?

Enter your answer as a decimal

I don't know

Anna and Louis are newlyweds and plan to have children.

Suppose they have three children. Let X be the number of boys, where $X=0,1,2,3$ .

*Assume the probability of getting a boy is 50% and the probability of getting a girl is 50%.

Complete the probability table:

For P(X), provide answers with three decimal places (e.g. 0.175)

Illustration for practice question: Randomness and Probability

X	0	1	2	3
P(X=x)

I don't know

Practice: Expected Value & Standard Deviation

Find the expected value and standard deviation.

(i) Expected Value (Mean) [Provide your answer rounded to 1 decimal place](ii) Standard Deviation [Provide your answer rounded to 2 decimal places e.g. 0.37]

I don't know

Extra Practice

Probability

(iii) Calculate the expected number of 'Heads' and the standard deviation.

Discrete vs. Continuous

Which of the following is true?

Probability

(v) Suppose it costs $10 to play. The prize is $20. What is the expected net profit from a single play?

Randomness and probability

You are one of three players remaining in a poker tournament. The total prize is $1,000 where the winner gets $750 and the player who gets second place wins $250. The third-place finisher wins nothing.

Currently, you have the most chips ($$$). Player A has the least amount of chips ($). You are dealt with ace-queen suited. The other two players went all-in with pairs! It is your turn.

If you go on all-in, the pot will be so massive that whoever wins this crucial round will win the tournament. The photo shows that, if you go all-in (or simply call), you have a 27.91% chance of winning this round and knocking both players out - thus winning the tournament! But if you go all-in and lose this round, you will lose a lot of chips and will have no chance in winning the tournament. Instead, you will settle for second or third place, depending if another player gets knocked out or not after this round. If Player A wins this round, all three players remain but Player A will be guaranteed to win the tournament, forcing you and Player B to fight for second place. If Player B wins this round, Player A will be knocked out and you will settle for second place.

Probability

Henry is a server at a high end restaurant. X denotes how many times he makes a mistake with an order in a given day. Which of the following is true?

Randomness and Probability

Anna and Louis are newlyweds and plan to have children.

Suppose they have three children. Let X be the number of boys, where

X=0,1,2,3. Complete the probability table:

Randomness and Probability

Amori is good at math and spelling so she entered a math contest and a spelling bee. The

probability of her winning each contest is listed below. Assume the two contests are

independent.

Probability Density Function

(a) For the following probability density function, the expected value is 36.45. Determine the value of $a$ .

(b) Find the mean, variance, and standard deviation of the discrete random variable with the given PDF:

Mean:

Randomness and probability

A bag has three red and four black balls. One ball is drawn at a time, without replacement until a black ball is drawn. Let X represent the number of balls drawn until you get a black one.

Determine the average number of balls drawn if this experiment were repeated many times. Enter answer in one decimal place X.X (e.g. 2.5)

Randomness and Probability

A bag has three red and four black balls. One ball is drawn at a time, without replacement until a black ball is drawn. Let X represent the number of balls drawn until you get a black one.

Find the probability P(X) of each outcome X. Express P(X) in fractions (e.g. 5/8).

Randomness and Probability

The exact number of cookies in a box varies from box to box according to the probability distribution function (PDF) given by:

Provide all your answers with at least 2 decimal places (e.g. 0.38)

Randomness and Probability

Find the expected value and standard deviation.

Randomness and Probability

Which of the following are discrete random variables? (Check all that applies.)

Randomness and Probability

Suppose that the random variable $X$ is the number of complaint calls that a local computer repair store may receive during a day. The probability model of $X$ is shown below

$\begin{array}{|c|c|c|c|c|c|} \hline x&0&1&2&3&4\\ \hline P(X=x)&0.10&0.10&0.20&0.50&0.10\\ \hline \end{array}$

Given that the expected value of $X$ is 2.4, what is the standard deviation of $X$ ?

Randomness and Probability

Suppose that certain car parts are manufactured so that their heights are independent of each other, with a mean of $\mu=31.2$ cm and a standard deviation of $\sigma=0.1$ cm.

If 2 of these parts are stacked on top of each other to form a new part, what are the mean and standard deviation of this new part?

Randomness and Probability: Independent RV

Let X and Y be two independent random variables where $𝐸(𝑋) = 2, \ 𝐸(𝑌) = 5, \ 𝑉(𝑋) = 4,\ 𝑉(𝑌) = 1$ .

Randomness and Probability

Let X and Y be discrete random variables where $𝑌 = 5𝑋 + 1, 𝐸(𝑋) = 4$ and $𝑉(𝑋) = 2$

Randomness and Probability

Let B be a discrete random variable with $𝐸(𝐵) = 6.$ Find $𝐸(−3𝐵 − 1)$

Randomness and Probability

Let X be a discrete random variable where $𝐸(𝑋 + 2) = 8.$ Find $𝐸(𝑋)$

Randomness and Probability

Tony plays a game in which he draws one card. If the card is a heart, he wins $2. Otherwise, he loses $2.

Let X be the amount he will win. Find the standard deviation of X (that is, $\sigma_x$ ). (round to 3 decimal places)

Randomness and Probability

A biased coin is three times as likely to turn up heads as tails. The coin is tossed twice. Let $X$ be the number of heads obtained. Which of the following is the probability distribution of $X$ ?

Randomness and Probability

A certain game wheel consists of 10 equal size slices. 2 of those slices are red, 5 if those slices are blue, and 3 of those slices are green. A player spins the game wheel, and if it lands on red, they lose $10, if it lands on blue they don't win or lose any money, and if it lands on green, they win $6. What is the expected value for a player's winnings?

Randomness and Probability

Let y be the number of cars in a sample of 3 that need to have their cylinder replaced over the next eight years. Here is a probability table that shows the values of y and each individual probability

Find the expected number of cars that will need to have their cylinder replaced

Randomness and Probability: Expected Value

Randomness and Probability

Suppose that certain car parts are manufactured so that their heights are independent of each other, with a mean of $\mu=31.2$ cm and a standard deviation of $\sigma=0.1$ cm.

If 2 of these parts are stacked on top of each other to form a new part, what are the mean and standard deviation of this new part?

Outcome (X)	Win Neither	Win Math Only	Win Spelling Only	Win Both
X
P(X=x)

Randomness and Probability

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Discrete Random Variables

Practice: Discrete Random Variables

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Probability Distribution Function

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Expected Value, Variance, and Standard Deviation of Discrete Random Variables

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Example: Expected Value & Standard Deviation

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Example: Probability Distribution Function

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Practice: Expected Value & Standard Deviation

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Extra Practice