What Are Random Variables?

A random variable, typically denoted as X, is a variable whose possible values are outcomes of a random process.

There are two types of random variables: discrete and continuous.

Discrete Random Variables

A discrete random variable is a variable 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 times a coin lands on tails after being flipped 20 times.
The number of times a dice lands on the number 4 after being rolled 100 times.
The number of defective widgets in a box of 50 widgets.

A probability distribution for a discrete random variable tells us the probability that the random variable takes on certain values.

For example, suppose we roll a fair die one time. If we let X denote the probability that the die lands on a certain number, then the probability distribution can be written as:

P(X=1): 1/6
P(X=2): 1/6
P(X=3): 1/6
P(X=4): 1/6
P(X=5): 1/6
P(X=6): 1/6

Note:

For a probability distribution to be valid, it must satisfy the following two criteria:

1. The probability for each outcome must be between 0 and 1.

2. The sum of all of the probabilities must add up to 1.

Notice that the probability distribution for the die roll satisfies both of these criteria:

1. The probability for each outcome is between 0 and 1.

2. The sum of all of the probabilities add up to 1.

We can use a histogram to visualize the probability distribution:

PDF for a discrete random variable

A cumulative probability distribution for a discrete random variable tells us the probability that the variable takes on a value equal to or less than some value.

For example, the cumulative probability distribution for a die roll would look like:

P(X≤1): 1/6
P(X≤2): 2/6
P(X≤3): 3/6
P(X≤4): 4/6
P(X≤5): 5/6
P(X≤6): 6/6

The probability that the die lands on a one or less is simply 1/6, since it can’t land on a number less than one.

The probability that it lands on a two or less is P(X=1) + P(X=2) = 1/6 + 1/6 = 2/6.

Similarly, the probability that it lands on a three or less is P(X=1) + P(X=2) + P(X=3) = 1/6 + 1/6 + 1/6 = 3/6, and so on.

We can also use a histogram to visualize the cumulative probability distribution:

CDF for a discrete random variable

Continuous Random Variables

A continuous random variable is a variable which can take on an infinite number of possible values. Some examples of continuous random variables include:

Weight of an animal
Height of a person
Time required to run a marathon

For example, the height of a person could be 60.2 inches, 65.2344 inches, 70.431222 inches, etc. There are an infinite amount of possible values for height.

Rule of Thumb:

If you can count the number of outcomes, then you are working with a discrete random variable – e.g. counting the number of times a coin lands on heads.

But if you can measure the outcome, you are working with a continuous random variable – e.g. measuring height, weight, time, etc.

A probability distribution for a continuous random variable tells us the probability that the random variable takes on certain values.

However, unlike a probability distribution for discrete random variables, a probability distribution for a continuous random variable can only be used to tell us the probability that the variable takes on a range of values.

For example, suppose we want to know the probability that a burger from a particular restaurant weighs a quarter-pound (0.25 lbs). Since weight is a continuous variable, it can take on an infinite number of values.

For example, a given burger might actually weight 0.250001 pounds, or 0.24 pounds, or 0.2488 pounds. The probability that a given burger weights exactly .25 pounds is essentially zero.

Thus, we could only use a probability distribution to tell us the probability that a burger weighs less than 0.25 lbs, more than 0.25 lbs, or between some range (e.g between .23 lbs and .27 lbs).