Question: There are two coins, one with probability p1 of Heads and the other with probability p2 of Heads. One of the coins is randomly chosen (with equal probabilities for the two coins). It is then flipped n ≥ 2 times. Let X be the number of times it lands Heads.
(a) Find the PMF of X.
(b) What is the distribution of X if p1 = p2?
(c) Give an intuitive explanation of why X is not Binomial for p1 ≠ p2 (its distribution is called a mixture of two Binomials). You can assume that n is large for your explanation, so that the frequentist interpretation of probability can be applied.