Finding & Interpreting the Expected Value of a Continuous Random Variable

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  • 0:04 Random Variables Defined
  • 2:09 Probability Density Function
  • 2:42 Computation & Interpretation
  • 4:13 Lesson Summary
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Lesson Transcript
Instructor: Artem Cheprasov
How can you find the expected value of something like height distributions? This lesson explains how to find and interpret the expected value of a continuous random variable.

Random Variables Defined

Random processes are an inherent part of our world. Think about flipping a coin. Since the process is random by its very nature, there is no way to determine the outcome of a future coin flip. However, we can assign probabilities to future events of random processes. The mathematical construct that we utilize to achieve this goal is called a random variable.

In this lesson, you'll learn about the mathematical treatment of random processes and how to find and interpret the expected value of a continuous random variable.

Random variables designate the possible outcomes of random processes. They come in two types: discrete and continuous. A discrete random variable is associated with processes such as rolling a die and flipping a coin, in which there is a countable number of outcomes.

For example, we can define a random variable, Z, associated with rolling two 8-sided dice, as follows:


The random variable can take on the shown values because the lowest possible outcome is rolling a 1 on both dice, while the highest possible outcome is rolling an 8 on both dice.

On the other hand, a continuous random variable involves processes such as height and weight measurements, in which there is an infinite spectrum of possible outcomes. For example, the height of a person can be measured as 175.12 centimeters, but a more precise measurement may yield 175.117 centimeters, and an even more precise one may be 175.1168 centimeters, and so forth!

As an example, assume that the heights can range between 140 cm and 210 cm. Mathematically, the corresponding continuous random variable, X, would be written as an interval:


In other words, a continuous random variable does not have a countable number of possible outcomes.

Probability Density Function

The probability density function, which is another name for a continuous probability distribution function, is a graph of the probabilities associated with all the possible values a continuous random variable can take on. It appears as a continuous curve, with the random variable values plotted on the x-axis and their corresponding probabilities on the y-axis.

As shown in the Height Distribution graph, there is a continuous range of values between 140 cm and 200 cm.


Expected Value Computation and Interpretation

A useful quantity that we can compute for a continuous probability distribution is the expected value. It is the outcome that we should expect to obtain on average.

The expected value of a continuous random variable can be computed by integrating the product of the probability density function with x. Mathematically, it is defined as follows:


We integrate over the interval in which f(x) is not equal to zero. As a simple example, assume that f(x) is defined as follows:


We can solve for the expected value, E(x), as shown on the screen:


By plugging and chugging away, our answer becomes 18.7.

Referring back to the height distribution example, let's assume that f(x) is a Gaussian function with a mean, mu, of 170 and a standard deviation, sigma, of 10, as shown here:


We can set up the integral to compute the expected value as follows:


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