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(X₁ = 0) = 0.5³ = 0.125
P(X₁ = 1) = 3*0.5³ = 0.375
P(X₁ = 2) = 3*0.5³ = 0.375
P(X₁ = 3) = 0.5³ = 0.125

Similarly for the second person, we get:

P(X₂ = 0) = 0.5³ = 0.125
P(X₂ = 1) = 3*0.5³ = 0.375
P(X₂ = 2) = 3*0.5³ = 0.375
P(X₂ = 3) = 0.5³ = 0.125

The distribution of the sample mean is thus computed as:

P(Y = 0) = 0.125² = 0.015625
P(Y = 0.5) = 0.125*0.375*2 = 0.09375
P(Y = 1) = 0.375² + 2*0.125*0.375 = 0.234375
P(Y = 1.5) = 2*0.375² + 2*0.125² = 0.3125
P(Y = 2) = 0.375² = 0.140625
P(Y = 2.5) = 2*0.125*0.375 = 0.09375
P(Y = 3) = 0.125² = 0.015625