Question

Let {?1,?2, … , ?? } be ? independent random draws from any given distribution with finite expected value ? and variance ? 2 > 0. Let ?̅ = 1 ? ∑ ?? ? ?=1 denote the average draw, which in turn is a random variable with its own distribution. This question works through successive proofs to derive the expected value and variance of this distribution, culminating in a proof of the Law of Large Numbers.

a. Show that ?(?? + ??) = ??(?) + ??(?) for any random variables ? and ? and constants ? and ?, and use this result to show that ?(?̅) = ?.

b. Show that ???(?) = ?(? 2 ) − ?(?) 2 and hence ???(??) = ? 2???(?) for any random variable ? and constant ?.

c. Show that ?(??) = ?(?)?(?) and hence ???(? + ?) = ???(?) + ???(?) for any independent random variables ? and ?.

d. Show that ???(?̅) = ? 2 ? . e. Show that ?̅ → ? as ? → ∞, which is the Law of Large Numbers.

Answer 1