Understanding Independence of Random Variables in the Discrete Case

 Show in the discrete case that if X and Y are independent, then E[X|Y = y] = E[X] for all y.

In probability and statistics, we often deal with random variables, which are variables that take on a set of possible values with a certain probability. Independence between random variables X and Y is a property that states that the occurrence of one event has no effect on the occurrence of the other. In this blog post, we will demonstrate how independence between two discrete random variables X and Y leads to the result that E[X|Y=y] = E[X] for all y.

Expected Value of Independent Random Variables:

The expected value of a random variable X is denoted as E[X] and represents the average or expected outcome of a random event. It is calculated as the sum of all possible values of X multiplied by their respective probabilities. For example, if X takes on the values {1,2,3} with probabilities {0.1,0.5,0.4}, then E[X] can be calculated as:

E[X] = 1*0.1 + 2*0.5 + 3*0.4 = 2.3

Conditional Expectation:

The conditional expectation of a random variable X given an event Y is denoted as E[X|Y=y] and represents the expected value of X given that event Y has occurred. It is calculated as the sum of all possible values of X multiplied by their respective probabilities, conditioned on Y=y. For example, if X takes on the values {1,2,3} with probabilities {0.1,0.5,0.4}, and Y=1, then E[X|Y=1] can be calculated as:

E[X|Y=1] = 1*0.1 + 2*0.5 + 3*0.4 = 2.3

Independence and Conditional Expectation:

Now, let's consider the case where X and Y are independent random variables. Independence between X and Y means that the occurrence of one event has no effect on the occurrence of the other. In other words, the probability of X given Y=y is the same as the probability of X. This means that:

P(X=x|Y=y) = P(X=x) for all x,y

Using this property, we can show that E[X|Y=y] = E[X] for all y. To see this, consider the following calculation:

E[X|Y=y] = Sum over x of x * P(X=x|Y=y)

= Sum over x of x * P(X=x) (since X and Y are independent)

= E[X]

Therefore, we have shown that if X and Y are independent, then E[X|Y=y] = E[X] for all y in the discrete case.

In this blog post, we have demonstrated how independence between two discrete random variables X and Y leads to the result that E[X|Y=y] = E[X] for all y. This result is important in understanding the relationship between independence and conditional expectations and can be applied in various statistical and probabilistic settings.