Question: (8 points) Suppose that X ₁ and X ₂ 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

1. What are the possible values of \(\bar{X}\) = (X ₁ + X ₂ )/2?
2. Determine the probabilities that \(\bar{X}\) assumes the values in (a).
3. Using (b), directly compute E[ \(\bar{X}\) ] and var( \(\bar{X}\) ).
4. Are your answers to (c) consistent with the general formula that E[ \(\bar{X}\) ] = \(\mu \) and var( \(\bar{X}\) ) = σ ² /n, where \(\mu \) := E[X] and σ ² := var(X).

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