Question

Let X, X, X,...X, be a random sample from a distribution with mean and standard deviation μ,& respectively. Show that S² = ΣΧ-Χ) is unbiased

n-1

estimator of variance

Answer 1

[AI-generated answer — Bot GPT 3.5]

To show that S² = Σ(X- X)² / (n-1) is an unbiased estimator of variance, we need to show that the expected value of S² is equal to the true population variance σ².

First, let's calculate the expected value of S²:

E[S²] = E[Σ(X- X)² / (n-1)]
= 1/(n-1) * E[Σ(X- X)²]
= 1/(n-1) * ΣE[(X- X)²]
= 1/(n-1) * nE[(X- X)²] (since the random variables in the sample are iid)
= 1/(n-1) * nVar(X) (since Var(X) = E[(X- X)²])
= 1 - 1/n * nσ² (since Var(X) = σ²)
= σ²

Therefore, the expected value of S² is equal to the true population variance, which means that S² is an unbiased estimator of variance.