A random spectator is to be selected from the audience of a basketball game and

Question

A random spectator is to be selected from the audience of a basketball game and given the chance to shoot 10 free throws. Let Y be the number of free throws made by the selected spectator and let P be the probability of making a free-throw by the selected spectator. Assume that P has a beta distribution with parameters and and that Y |P has the Binomial(10, P ) distribution (refer to the hierarchical models lecture notes).

1. Find values of and such that EY =4.

2. Find E[Y |P = 1/4] mathematically.

3. Find Var[Y |P = 1/4] mathematically.

Explanation / Answer

P has a prior distribution of Beta(a,b) (Lets term a = alpha and b = beta).

So Ep(P) = a/a+b

Now Y|P ~ Binomial(10,p) distribution. So E(Y|P) = 10*P and Var(Y|P) = 10*P*(1-P)

E(Y) = Ep(EY|P(Y)) = Ep(10P) = 10*a/a+b

Given E(Y) = 4.

So 10a/a+b = 4

=> 6a = 4b => 3a = 2b which means a=2 and b=3 since 2 and 3 are mutually prime.

So a=2 and b=3.

2. The E(Y|P) = n*P

Here n=10 and P is given to be 1/4.

So E(Y|P=1/4) = 10*1/4 = 2.5

3. Var(Y|P=1/4) = n*p*(1-p) = 10*1/4*(1-1/4) = 10 * 1/4 * 3/4 = 30/16 = 1.88

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