Top 10 Conditional Expectation Interview Questions and Answers

Conditional expectation is a key concept in probability theory and statistics that often comes up in quantitative interviews, especially for roles in finance, data science, and machine learning. Knowing how to calculate conditional expectations and interpret them is crucial.

In this article, I’ll go over the top 10 conditional expectation interview questions frequently asked, provide sample answers, and explain the intuition behind each one. Mastering these will ensure you ace the probability and stats portion of your next interview!

1. What is meant by conditional expectation and how is it calculated?

The conditional expectation E(X|Y) refers to the expected value of the random variable X given that another random variable Y has occurred or is known. Intuitively, it tells us the average value we expect X to take based on the specific value or realization of Y.

The general formula for conditional expectation is:

E(X|Y) = ∑_y E(X|Y=y)P(Y=y)

Where E(X|Y=y) is the expected value of X given a particular value y that Y takes, and P(Y=y) is the probability of Y taking that value y

So we calculate the conditional expectation by taking a weighted average of the expected values of X for each possible value of Y, weighted by the probabilities of those Y values occurring.

2. How do you calculate E(X|Y) when X and Y are discrete random variables?

When X and Y are both discrete, we can calculate the conditional expectation directly using the formula:

E(X|Y=y) = ∑_x xP(X=x|Y=y)

Where P(X=x|Y=y) is the conditional probability mass function of X given the value y of Y.

Then substitute this into the general conditional expectation formula:

E(X|Y) = ∑_y [ ∑_x xP(X=x|Y=y) ] P(Y=y)

Which is a double summation over all possible values of X and Y.

For example, if X ~ Binomial(10, 0.3) and Y ~ Binomial(10, 0.5), then:

E(X|Y=5) = 0P(X=0|Y=5) + 1P(X=1|Y=5) + … + 10*P(X=10|Y=5)

And repeat for other values of Y to get E(X|Y).

3. How do you interpret the conditional expectation of one random variable given another?

The conditional expectation E(X|Y) can be interpreted as the best predictor of X given knowledge of Y, or the average value we expect X to take based on the observed value of Y.

Some key properties:

• E(X|Y) is a function of Y, not a constant. As the value of Y changes, the conditional expectation will change.

• E(X|Y) is the minimum MSE predictor of X given Y. No other function of Y will give a lower MSE when predicting X.

• E(X|Y=y) will be equal to E(X) if X and Y are independent. The conditional expectation equals the unconditional expectation if knowing Y provides no additional information about X.

4. If Cov(X,Y) = 0, what can you say about E(X|Y)?

If the covariance between two random variables X and Y is 0, then X and Y are uncorrelated. When X and Y are uncorrelated, the conditional expectation E(X|Y) is simply equal to the unconditional expectation E(X):

E(X|Y) = E(X)

This makes intuitive sense – if X and Y are uncorrelated, then knowing the value of Y provides no additional information about the likely value of X. The best prediction of X is simply its mean, regardless of Y.

So when covariance is zero, the conditional expectation equals the unconditional expectation. This provides a quick way to calculate E(X|Y) when you know X and Y are uncorrelated.

5. How do you calculate the conditional expectation of a normal random variable?

For multivariate normal random variables, the conditional expectation takes on a very specific linear form.

Let X and Y be jointly normal with means μX, μY, variances σX^2, σY^2 , and correlation ρ. Then:

E(X|Y) = μX + ρσX/σY (Y – μY)

The conditional expectation is a linear function of Y, with slope equal to the correlation ρ multiplied by the standard deviation ratio.

Intuitively, as Y increases, our expectation of X will increase linearly if X and Y are positively correlated. The slope represents how strongly X is influenced by movements in Y.

For normal variables, you can quickly write down the conditional expectation using this formula rather than evaluating integrals.

6. Suppose X and Y have a bivariate normal distribution. How do you calculate E(X|Y>k) for some constant k?

When X and Y are bivariate normal, we can use properties of truncated normals to calculate conditional expectations like E(X|Y>k).

Let φ(Y) and Φ(Y) represent the PDF and CDF of a standard normal variable. Then:

E(X|Y>k) = μX + ρσX/σY * [φ(k)/(1 – Φ(k))]

The intuition is we are truncating the joint normal distribution to only where Y > k. This changes the expected value of Y in that truncated region, which then affects E(X|Y) through their correlation ρ.

You can also calculate conditional expectations like E(X|Y<k) or E(X|a<Y<b) using similar logic with truncated normal CDFs and densities. The bivariate normal case allows for many nice closed form conditional expectation solutions.

7. What is the significance of conditional expectation in regression analysis?

In the linear regression model:

Y = β0 + β1X + ε

Where ε is the error term with mean 0, the conditional expectation E(Y|X) represents the regression function – the expected value of Y given X.

Specifically:

E(Y|X) = β0 + β1X

So the conditional expectation as a function of X gives the predicted value or fitted values from the regression. The regression line is the conditional mean.

This demonstrates the close connection between conditional expectation and regression. Regression analysis seeks to estimate E(Y|X) by finding the linear function of X that best predicts Y on average.

8. Suppose {Xt} is a martingale process. What can you say about E(Xt|Xt-1)?

For a martingale process, the conditional expectation of the next value given the current value is just equal to the current value:

E(Xt|Xt-1) = Xt-1

This property characterizes the martingale, and is sometimes used as an alternative definition. It means the best predictor of the next X is just the current X – the new value provides no new information.

Intuitively, the changes in a martingale are unpredictable and centered around 0. Knowing the past doesn’t help predict the future. So the conditional expectation is simply equal to the current level.

9. What is the conditioning property of conditional expectation?

An important property of conditional expectation is:

E(E(X|Y)|Y) = E(X|Y)

That is, the conditional expectation operator commutes with further conditioning. The conditional expectation of X given Y is itself a function of Y. If we take the conditional expectation of that given Y, we get the original E(X|Y).

This conditioning property is very useful for iteratively calculating conditional expectations. We can break them down into steps while preserving equality at each step.

10. How do you calculate E(g(X,Y)|X) for some function g?

For a general function g(X,Y), we can use the law of iterative expectations:

E(g(X,Y)|X) = E(E(g(X,Y)|X,Y)|X)

Break it down into two steps:

1. Calculate the inner conditional expectation E(g(X,Y)|X,Y). This holds X and Y fixed.

2. Take the expectation of that given just X.

Intuitively, we first find the conditional expectation for each combination of X and Y values. We then average those results based on the marginal distribution of X.

This allows calculating complex conditional expectations by iteratively conditioning on subsets of the variables.