A **random variable**, usually denoted as X, is a variable whose values are numerical outcomes of some random process. There are two types of random variables: discrete and continuous. This section will focus on discrete random variables.

**Discrete Random Variables**

A **discrete random variable** is one which can take on only a countable number of distinct values like 0, 1, 2, 3, 4, 5…100, 1 million, etc. Some examples of discrete random variables include:

The number of members in a family.

The number of football players on a team.

The amount of animals at a zoo.

The number of chairs at a table.

The number of patients sitting in a doctor’s office.

**Probability Distributions**

The **probability distribution** of a random variable is a list of probabilities associated with each of its possible values.

For example, suppose we flip a coin three times. Let random variable X = the number of times the coin lands on heads. Assuming “H” stands for “heads” and “T” stands for “tails”, here are the eight possible outcomes of three coin flips:

**HHH HHT HTH HTT THH THT TTH TTT**

To make a probability distribution for X, we need to identify all the possible outcomes of X*. *From the outcomes listed above, we can see that the number of heads after three flips could be: 0, 1, 2, or 3.

Next, we need to find the probability of each outcome.

The probability that the coin lands on heads 0 times is: P(X = 0) = 1/8

**HHH HHT HTH HTT THH THT TTH TTT**

The probability that the coin lands on heads 1 time is: P(X = 1) = 3/8

**HHH HHT HTH HTT THH THT TTH TTT**

The probability that the coin lands on heads 2 times is: P(X = 2) = 3/8

**HHH HHT HTH HTT THH THT TTH TTT**

And the probability that the coin lands on heads 3 times is: P(X = 3) = 1/8

**HHH HHT HTH HTT THH THT TTH TTT**

So the probability distribution of X is:

And we can visual this probability distribution in a plot with the value for X on the x-axis and the probability on the y-axis:

**Valid Discrete Probability Distributions**

A discrete probability distribution is *valid *if the probabilities for all of the possible values adds up to one. The probability distribution in our previous example is valid because all of the probabilities add up to one:

A discrete probability distribution is *not *valid if the probabilities add up to anything but one. For example, the following two probability distributions are not valid:

**Finding Probabilities of Discrete Random Variables**

We can use probability distributions to answer questions about probability. Consider again our probability distribution of a coin landing on heads during three flips:

**What is the probability that a coin doesn’t land on heads during three flips?**

The probability that a coin doesn’t land on heads during three flips is P(X=0) = 1/8.

**What is the probability that a coin lands on heads at least once during three flips?**

To find this probability, we can add up the probabilities that the coin lands on heads once, twice, or three times: P(X≥ 1) = 3/8 + 3/8 + 1/8 = **7/8**.

**What is the probability that a coin lands on heads less than twice during three flips?**

To find this probability, we can add up the probabilities that the coin lands on heads zero times or one time: P(X < 2) = 1/8 + 3/8 = 4/8 =** 1/2**.

**The Mean & Variance of a Discrete Random Variable**

To find the mean (or the “expected value”) of a discrete random variable, denoted as E(X), simply multiply each possible value with its probability, then take the sum:

Expected number of heads = (0*1/8) + (1*3/8) + (2*3/8) + (3*1/8) = 0 + 3/8 + 6/8 + 3/8 = 12/8 = **1.5**

The expected number of heads from three coin flips is **1.5**.

To find the variance of a discrete random variable, denoted as Var(X), we subtract the expected value from each possible value, square this number, then multiply by its probability. Then we sum all these numbers.

Var(X) = (0-1.5)^{2} *1/8 + (1-1.5)^{2} * 3/8 + (2-1.5)^{2} * 3/8 + (3-1.5)^{2} * 1/8 = **0.75**

To find the standard deviation of X, we simply take the square root of the variance: √0.75 = **0.866**