You have 3 coins that look identical, but one is not a fair coin. The probability the unfair coin show heads when tossed is 3/4, the other two coins are fair and have probability 1/2 of showing heads when tossed.
You pick one of three coins uniformly at random and toss it n times, (where n is an arbitrary fixed positive integer).
Let Y be the number of times the coin shows heads. Let X be the probability the coin you choose shows heads when flipped. (Note that X is a random variable because you’re randomly choosing a coin).
a) What is the pmf of X?
b) For each x that the random variable X can equal, give the conditional distribution of Y given X = x.
c) Determine the unconditional distribution of Y
d) Compute the expectation of Y