Expected Value & Variance
of Continuous Random Variable

The expected value (mean) and variance are two useful summaries because they help us identify the middle and variability of a probability distribution.

And as we saw with discrete random variables, the mean of a continuous random variable is usually called the expected value. Note that the standard deviation is sometimes called the standard error.

Formulas

What is so unique is that the formulas for finding the mean, variance, and standard deviation of a continuous random variable is almost identical to how we find the mean and variance for a discrete random variable as discussed on the probability course. The only difference is integration!

Worked Example

So, now that we are armed with our formulas for finding the measure of center and the measure of dispersion let’s see them in action.

For example, let’s determine the expected value and variance of the probability distribution over the specified range.

And now that we know that the mean is 2/5, we can find the variance and standard deviation.

And that’s it!

I will be reminding you of your integration skills like u-substitution, integration by parts, and improper integrals along the way, so you’ll never get stuck or confused. We’ll work through each example step-by-step.

So, let’s jump right in and use our formulas to successfully calculate the expected value, variance, and standard deviation for continuous distributions.

Expected Value Variance Continuous Random Variable – Lesson & Examples (Video)

1 hr 25 min

• Introduction to Video: Mean and Variance for Continuous Random Variables
• 00:00:28 – Properties and formulas for mean and variance of continuous random variables
• Exclusive Content for Members Only

• 00:07:29 – Find the mean and variance of a discrete random variable (Example #1)
• 00:12:06 – Find the mean and variance of a continuous random variable (Example #2)
• 00:20:01 – Determine the mean and variance (Example #3)
• 00:30:18 – Determine the mean of a discrete random variable (Example #4)
• 00:33:39 – Find the mean of the continuous random variable (Example #5)
• 00:44:04 – Given a triangular probability density function find the pdf formula (Example #6a)
• 00:49:58 – Using the pdf formula from part a, find the mean (Example #6b)
• 00:56:41 – Find the probability of the continuous distribution (Example #6c)
• 01:02:44 – Using various integration techniques, find the expected value and variance (Example #7)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions