Solving Conditional Expectations of Joint Probabilities with X, Y, Z

Suppose p(x, y, z), the joint probability mass function of the random variables X,

Y, and Z, is given by

p(1, 1, 1) = 1/8 , p(2, 1, 1) = 1/4 ,

p(1, 1, 2) = 1/8 , p(2, 1, 2) = 3/16 ,

p(1, 2, 1) = 1/16 , p(2, 2, 1) = 0,

p(1, 2, 2) = 0, p(2, 2, 2) = 1/4 , What is E[X|Y = 2]? What is E[X|Y = 2, Z = 1]?

In probability and statistics, the conditional expectation is the expected value of a random variable given specific conditions. In this blog post, we will solve the conditional expectations of X, Y, and Z, given their joint probability mass function.

Calculating E[X|Y = 2]:

To calculate the expected value of X given Y=2, we need to calculate the weighted average of the values of X, with their respective probabilities.

p(1, 2, 1) = 1/16, p(2, 2, 1) = 0

p(1, 2, 2) = 0, p(2, 2, 2) = 1/4

The expected value of X given Y=2 is 1/42 + 1/41 = 3/4.

Calculating E[X|Y = 2, Z = 1]:

To calculate the expected value of X given Y=2 and Z=1, we need to calculate the weighted average of the values of X, given their respective probabilities.

p(1, 2, 1) = 1/16, p(2, 2, 1) = 0

The expected value of X given Y=2 and Z=1 is 1/16*1 + 0 = 1/16.

We have solved the conditional expectations of X, Y, and Z, given their joint probability mass function. We have calculated the expected value of X given Y=2 and the expected value of X given Y=2 and Z=1. Understanding conditional expectations is important in solving real-world problems, as it helps us make predictions and decisions based on the given conditions.