Question

Suppose two people flip a coin three times. Let X1, X2 denote the number of tails flipped by the first and second person. Find the sampling distribution of the sample mean

PLEASE EXPLAIN, I WILL GIVE A THUMBS UP FOR A NICE EXPLANATION. THANK YOU IN ADVANCE :)

Answer #1

For each of the 2 people,

P(X_{1} = 0) = 0.5^{3} = 0.125

P(X_{1} = 1) = 3*0.5^{3} = 0.375

P(X_{1} = 2) = 3*0.5^{3} = 0.375

P(X_{1} = 3) = 0.5^{3} = 0.125

Similarly for the second person, we get:

P(X_{2} = 0) = 0.5^{3} = 0.125

P(X_{2} = 1) = 3*0.5^{3} = 0.375

P(X_{2} = 2) = 3*0.5^{3} = 0.375

P(X_{2} = 3) = 0.5^{3} = 0.125

The distribution of the sample mean is thus computed as:

P(Y = 0) = 0.125^{2} = 0.015625

P(Y = 0.5) = 0.125*0.375*2 = 0.09375

P(Y = 1) = 0.375^{2} + 2*0.125*0.375 = 0.234375

P(Y = 1.5) = 2*0.375^{2} + 2*0.125^{2} =
0.3125

P(Y = 2) = 0.375^{2} = 0.140625

P(Y = 2.5) = 2*0.125*0.375 = 0.09375

P(Y = 3) = 0.125^{2} = 0.015625

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