- #1

maverick280857

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I have a question regarding i.i.d. random variables. Suppose [itex]X_1,X_2,\ldots[/itex] is sequence of independent and identically distributed random variables with probability density function [itex]f_{X}(x)[/itex], mean = [itex]\mu[/itex] and variance = [itex]\sigma^2 < \infty[/itex].

Define

[tex]Y_{n} = \frac{1}{n}\sum_{i=1}^{n}X_{i}[/tex]

Without knowing the form of [itex]f_{X}[/itex], how does one prove that [itex]var(Y_{n}) = \sigma^2/n[/itex]?

I suppose this is a standard theorem/result, but any hints/ideas to prove this would be appreciated.

Thanks.