Statistics How To

Conditional Expectation: Definition & Step by Step Example

Probability >

You may want to read this other article first: What is Conditional Probability?

The conditional expectation (also called the conditional mean or conditional expected value) is simply the mean, calculated after a set of prior conditions has happened. Put more formally, the conditional expectation, E[X|Y], of a random variable is that variable’s expected value, calculated with respect to its conditional probability distribution. You can also say that E[X|Y] is the function of Y that is the best approximation for X (or, equally, the function of X that is the best approximation for Y).

Formula and Worked Example

Suppose we have two discrete random variables X and Y. with x ∈ Range(X), the condition expectation of Y given X = x:

Note: X given Y = y is defined in the same way (just switch the variables).

The formula might look a little daunting, but it’s actually pretty simple to work. What it is telling you to do is find the proportions of the “conditional” part (all the values where X = 1), multiply those by the Y values, then sum them all up. The process becomes much simpler if you create a joint distribution table.

Question: What is E(Y |X = 1) — the conditional expectation of Y, given that X = 1?

Step 1: Find the sum of the “given” value (X = 1). This is already given in the total column of our table:
0.03 + 0.15 + 0.15 + 0.16 = 0.49.

Step 2: Divide each value in the X = 1 column by the total from Step 1:

  • 0.03 / 0.49 = 0.061
  • 0.15 / 0.49 = 0.306
  • 0.15 / 0.49 = 0.306
  • 0.16 / 0.49 = 0.327

Step 3: Multiply each answer from Step 2 by the corresponding Y value (in the left-hand column):
0.0612244898 * -1 = -0.061
0.306122449 * 2.75 = 0.842
0.306122449 * 3 = 0.918
0.3265306122 * 4.55 = 1.486

Step 3: Sum the values in Step 2:
E(Y|X = 1) = -0.061 + 0.842 + 0.918 + 1.486 = 3.19

E(Y|X = 1) = 3.19

Continuous Case

For continuous distributions, expectations must first be defined by a limiting process. The result is a function of y and x that you can interpret as a random variable. Basically, an integral represents the limiting process and replaces the sums from the example above. The formula becomes:

E(Y∣X = x) = ∫t yh (y|x) dy

When we are dealing with continuous random variables, we don’t have the individual probabilities for each x that we had in the random variable example above. Instead, what you have is a probability density function for each individual x-value. To get the expected value, you integrate these pdfs over a tiny interval to essentially force the pdf to give you an approximate probability. Then, as in the steps above, you sum everything up.


If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.

Comments are now closed for this post. Need help or want to post a correction? Please post a comment on our Facebook page and I'll do my best to help!
Conditional Expectation: Definition & Step by Step Example was last modified: October 12th, 2017 by Stephanie