Lesson Explainer: Continuous Random Variables Mathematics

In this explainer, we will learn how to describe the probability density function of a continuous random variable and use it to find the probability for some event.

Continuous random variables take an infinite number of real number values in a continuum. The probability of a continuous random variable taking a particular value is zero: that is, for any value . The fact that the random variable has a zero probability of taking any one value is what distinguishes the continuous random variables from the discrete ones.

When working with a continuous random variable, boundary conditions of events can be neglected. In other words, strict and weak inequalities, and , describing different events are interchangeable. To see why this is the case, let us examine the probability for some real number . Since the events and are mutually exclusive, we have

But because for the continuous random variable , we have the equivalency . Similarly, for any upper bound and lower bound , we have the identity

A continuous random variable is characterized by its probability density function, which is a nonnegative function whose total area under the curve is 1. The area under the curve of the probability density function represents the probability of the whole sample space. We recall the rule of probability, which states that the probabilities of mutually exclusive events add up to 1. This rule necessitates the total area under the curve to be 1.

Consider a probability density function whose graph is given below.

We note that this function is never negative, and the total area under the curve is equal to 1. So this graph is a probability density function by the definition above.

When a probability density function contains an unknown constant, we can often identify the unknown constant using one of the conditions in the definition above. Any probability function satisfies the identity

Based on our previous discussion, we recall that this identity roots from the rule of probability.

Let us consider a few examples where we use the rule of probability to identify unknown constants within probability density functions.

Let us consider another example to apply the Rule of Probability to compute an unknown constant in the probability density function.

Say that a continuous random variable has the probability density function in the following figure, and that is an interval. Then, the probability of an event for is equal to the area under the curve over the given interval . We recall that since is a nonnegative function, then the area under the curve is equal to the definite integral of over the interval . For instance, the probability for an upper bound is equal to the area under the curve over the interval , which is pictured below.

This is given by the integral

Likewise, to compute the probability for the upper and lower bounds and , we compute the area over the interval as pictured below.

This is given by the integral

In general, we have the following formulae.

Although the integral formulae above can always be used to compute probabilities, it may be more efficient sometimes to use geometry when possible. This is the case when the graph of the probability density function forms simple geometric shapes such as a triangle, a trapezoid, or a semicircle.

Let us consider an example where the graph of the probability density function is given as a trapezoid. In this example, we will use geometry to compute the probability.

Example 3: Computing the Probability of a Continuous Random Variable Using Graphs

Let be a continuous random variable with the probability density function represented by the following graph. Find .

Answer

We are given the probability density function as a graph, so we begin by highlighting the region under the curve over the interval . In particular, we can deduce that the height at is because it is precisely halfway between 4 and 6.

We recall that the area of a trapezoid is given by the formula

The provided graph of our probability density function is a trapezoid with base , top , and height 1. So, the area of the trapezoid is

Then, the probability . We note that this is a reasonable answer for probability since is between 0 and 1.

When the graphs of probability density functions are not provided, it is often easier to use the integral formulae to compute the required probabilities. In the next two examples, we will use the given probability density functions with the integral formulae to compute the probabilities.

Let us consider another example using the integral formulae to become familiar with different contexts.