Overview
On this page, we discuss the concepts of expected value and variance of a continuous random variable.
Basic learning objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.

Use the definition of the expected value and the variance of a continuous random variable to calculate those values from a continuous random variable with a known density function.

Review the properties of expected value and variance that we showed for discrete random variables, and note that they apply in the continuous case as well.
Advanced learning objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:

Use the tail sum formula to calculate the expected value of a continuous random variable from its cumulative distribution function.

Understand the meaning of Chebyshev’s inequality, and know how to use it to bound the probability that any random variable takes values in a given interval.
To prepare for class

Watch the following video (by jbstatistics) which shows example calculations of the expectation and variance of a continuous random variable with a simple density function:

Watch the following video (by Brendan Cordy) which shows calculations of the expectation for more complicated density functions (in particular involving Integration by Parts and Improper Integrals):
After class

Watch the following video (by Anish Turlapaty) which explains the Tail Sum Formula in the case of a discrete random variable  which should give you an intuition on why a similar formula also works for continuous random variables:

Watch the following video (by Brendan Cordy) which explains Chebyshev’s inequality and how to use it:

Watch the following video (by Brendan Cordy) which shows various example calculation involving the variance and standard deviation of a continuous random variable (and some other things):

Review general facts about integration techniques as needed using these Calculus 2 review pages: