Question: Suppose that X₁ and X₂ constitute a sample of size 2 from a population
in which a typical value X is equal to either 1 or 2 with respective probabilities

P(X =1) = 0.7 P(X =2) = 0.3

(a) What are the possible values of \(\bar{X}\) = (X₁ + X₂)/2?

(b) Determine the probabilities that \(\bar{X}\) assumes the values in (a).

(c) Using (b), directly compute E[ \(\bar{X}\) ] and var( \(\bar{X}\) ).

(d) Are your answers to (c) consistent with the general formula that E[ \(\bar{X}\) ] = \(\mu \) and var( \(\bar{X}\) ) = σ²/n, where \(\mu \) := E[X] and σ² := var(X).