Joe has two coins, A and B, in front of him. The probability of Heads at...

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Joe has two coins, A and B, in front of him. The probability of Heads at each toss is p = 0.5 for coin A and q = 0.9 for coin B. Joe chooses one of the two coins at random (both choices are equally likely). He then continues with 5 tosses of the chosen coin; these tosses are conditionally independent given the choice of the coin. Let: H, : the event that Joe's ith coin toss resulted in Heads; N: the number of Heads in Joe's coin tosses. 1. For i € {0, 1,..,5}. pN (i), the pmf of N, is of the form 1 2. Find E[N]. E[N] = 3. Find the conditional variance of N, in a conditional model where we condition on having chosen coin A and the first two tosses resulting in Heads. Var(N | A, H1, H2) = 4. Are the events H1 and {N = 5} independent? Select an option • 5. Given that the 3rd toss resulted in Heads, what is the probability that coin A was chosen? Joe has two coins, A and B, in front of him. The probability of Heads at each toss is p = 0.5 for coin A and q = 0.9 for coin B. Joe chooses one of the two coins at random (both choices are equally likely). He then continues with 5 tosses of the chosen coin; these tosses are conditionally independent given the choice of the coin. Let: H, : the event that Joe's ith coin toss resulted in Heads; N: the number of Heads in Joe's coin tosses. 1. For i € {0, 1,..,5}. pN (i), the pmf of N, is of the form 1 2. Find E[N]. E[N] = 3. Find the conditional variance of N, in a conditional model where we condition on having chosen coin A and the first two tosses resulting in Heads. Var(N | A, H1, H2) = 4. Are the events H1 and {N = 5} independent? Select an option • 5. Given that the 3rd toss resulted in Heads, what is the probability that coin A was chosen?

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