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Chapter 15: Q2P (page 775)

Let x1,x2,..,xnbe independent random variables, each with density function f(x), expected value渭 , and variance蟽2 . Define the sample meanby.x鈭�=鈭�i=1nxiShowthatE(x鈭�)=渭,and .var(x鈭�)=蟽2n (See Problems 5.9,5.13and6.15.)

Short Answer

Expert verified

The statement has been proven.

Step by step solution

01

Given Information

x1,x2,...,xn be the independent random variables, each with density functionf(x) .

02

Definition of the Arithmetic mean.

The arithmetic mean is the sum of all the values divided by the total number of values.

03

Prove the statement.

Let x1,x2,...,xnbe the independent random variable.

The mean is given below.

x1,x2,...,xnE(x炉)=E(1n鈭�i=1nxi)=1n鈭�i=1nE(xi)=1n鈰�n渭=渭

The variance is given below.

Var(x炉)=Var(1n鈭�i=1nxi)=1n2鈭�i=1nVar(xi)=1n2鈰�n蟽2=蟽2n

Hence the statement has been proven.

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