If X is a random variable that can take any value in {1, 2, 3, . . .}, and if A is an event, then the conditional expected value E(X | A) is defined as
E(X | A) =
k . Pr(X = k | A). ( this picture not opening is a symbol of Sigma with n=1 under it and infinity sign on its top)
In words, E(X | A) is the expected value of X, when you are given that the event A occurs. You roll a fair die repeatedly, and independently, until you see the number 6. Define the random variable X to be the number of times you roll the die (this includes the last roll, in which you see the number 6). We have seen in class that E(X) = 6. Let A be the event
A = “the results of all rolls are even numbers”.
Determine the conditional expected value E(X | A).
Hint: The answer is not what you expect. We have seen in class that k · x k−1 = 1/(1 − x) 2 . (( this picture not opening is Sigma with n=1 under it and infinity sign on its top)
