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 continuous random variables.
Continuous Random Variables
A continuous random variable is one which can take on an infinite number of possible values. Some examples of continuous random variables include:
Height of a person
Weight of an animal
Time required to run a mile
Amount of rainfall on a certain day
A continuous random variable is not defined at specific values. Instead, it’s defined over an interval of values and is represented by the area under a curve called a probability density function.
For example, suppose a restaurant advertises a burger that 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.
Imagine that we record the weight of 100 random burgers from this restaurant and make a probability density histogram of the resulting weights:
More on Probability Density Functions
The total area under the curve is equal to one.
We can use a probability density function to find the probability that a random variable takes on a value within an interval, but we cannot use it to find the probability that a random variable takes on a specific value. For example, we can use the curve above to find the probability that a burger weighs between 0.23 and 0.27 pounds, but we cannot use it to find the probability that a burger weights exactly 0.25 pounds.
The most common continuous probability distribution is the normal distribution, which is described in detail in the next section.