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Let X be a b (1, 1/2)RV, and consider all possible random samples of size 3...

Let X be a b (1, 1/2)RV, and consider all possible random samples of size 3 on X. Compute X bar and S2 for each of the eight samples, and compute the PMFs of X bar and s2

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Answer #1

Let X be Bernaulli random Variable with probability p = 1/2 ( i.e. Binomial random variables with n=1 and p=1/2).

The probability mass function of X is

Let Xbar and S2 be sample mean and sample variance respectively.

We have to draw sample of size 3 ( n=3)

The possible samples are ( 0,0,0) , ( 0, 0, 1) , ( 0, 1, 0), ( 0, 1, 1), ( 1, 0 ,0 ) , (1,1,0) ,(1, 0, 1) , (1,1,1).

Sample No. 1 2 3 Xbar S2
1 0 0 0 0 0.0000
2 0 0 1 0.3333 0.3333
3 0 1 0 0.3333 0.3333
4 0 1 1 0.6667 0.3333
5 1 0 0 0.3333 0.3333
6 1 0 1 0.6667 0.3333
7 1 1 0 0.6667 0.3333
8 1 1 1 1 0.0000

From table Xbar take 4 values 0, 0.3333, 0.6667, 1

and S2 take 2 values 0 and 0.3333.

The probability mass function ( PMF) of Xbar is

Xbar 0 0.3333 0.6667 1 Total
P(Xbar) 1/8 3/8 3/8 1/8 1

The probability mass function S2 is

S2 0 0.3333 Total
P(S2) 2/8 6/8 1
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