Question

A bag contains three balls numbered 1, 2, and 3. Two balls are sampled with replacement. Let X1 be the number on the first ball sampled and let X2 be the number on the second ball sampled. Find the probability mass function of the sample mean (X1 + X2)/2.

Answer 1

The probability of drawing 1 ball(Any numbered) out of 3 is 1/3

P(X) = 1/3  

Case1 Here X1 = X2

So consider both balls are numbered 1 so Probability of this happening will be 1/3*1/3 = 1/9

P(1,1) = 1/9

similarly, P(2,2) = 1/9 P(3,3) = 1/9

Mean will be the same of 2 equal no.

Case2 Here X1 X2

So consider 1 ball numbered 1 and other ball numbered 2 and for this, there will be 2 possibilities (1,2 and 2,1)

So P(1,2 or 2,1 ) = 2 * 1/3 * 1/3 = 2/9

Similar Prob will be for all combinations where x1 and x2 are unequal

So now X = (X1 + X2)/2

P(1) = 1/9 Only one case P(1,1)

P(1.5) = 2/9 P(1,2 or 2,1)

P(2) = 1/9 + 2/9 = 1/3 P(1,2 or 2,1) or P(2,2)

P(2.5) = 2/9 P(2,3 or 3,2)

P(3) = 1/9 P(3,3)

If any doubt please feel free to ask through the comments section and if no doubt please give a like. Thank You