Fall 2011 Strategic Practice 10: Section 1 (Conditional Expectation and Conditional Variance) - Question 5
One of two identical-looking coins is picked from a hat randomly, where one coin has probability $p_{1}$ of Heads and the other has probability $p_{2}$ of Heads. Let X be the number of Heads after flipping the chosen coin n times. Find the mean and variance of X.
Solution: The distribution of X is a mixture of two Binomials; we have seen before that X is not Binomial unless p1 = p2. Let I be the indicator of having the p1 coin. Apply Adam's Law and Eve's Law. Consult iTunes course for full solution.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."