Let X1, . . . , Xn are independent and identically distributed random variables with meanµ and variance σ2. Apply the result in Problem 4 for X = ∑ni=1 Xi/n to show the lawof large number
Question
Let X1, . . . , Xn are independent and identically distributed random variables with meanµ and variance σ2. Apply the result in Problem 4 for X = ∑ni=1 Xi/n to show the lawof large number
Solution
The Law of Large Numbers states that as the size of a sample increases, the sample mean will get closer and closer to the population mean.
Let's denote X_bar as the sample mean, which is X_bar = ∑ from i=1 to n of Xi/n.
Since X1, . . . , Xn are independent and identically distributed (i.i.d.) random variables, the expected value of the sample mean is the population mean:
E(X_bar) = E(∑ from i=1 to n of Xi/n) = ∑ from i=1 to n of E(Xi)/n = n*µ/n = µ.
The variance of the sample mean is the population variance divided by the sample size:
Var(X_bar) = Var(∑ from i=1 to n of Xi/n) = ∑ from i=1 to n of Var(Xi)/n^2 = n*σ^2/n^2 = σ^2/n.
As n approaches infinity, the variance of the sample mean approaches 0:
lim as n->∞ of Var(X_bar) = lim as n->∞ of σ^2/n = 0.
This means that for large n, X_bar is approximately normally distributed (by the Central Limit Theorem) with mean µ and variance 0. In other words, X_bar is approximately equal to µ with high probability. This is the Law of Large Numbers.
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