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Chapter 15: Q2P (page 775)

Let x1,x2,..,xnbe independent random variables, each with density function f(x), expected valueμ , and varianceσ2 . Define the sample meanby.x−=∑i=1nxiShowthatE(x−)=μ,and .var(x−)=σ2n (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¯)=E(1n∑i=1nxi)=1n∑i=1nE(xi)=1n⋅nμ=μ

The variance is given below.

Var(x¯)=Var(1n∑i=1nxi)=1n2∑i=1nVar(xi)=1n2⋅nσ2=σ2n

Hence the statement has been proven.

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Most popular questions from this chapter

Define s by the equation.s2=(1/n)∑i=1n(xi−x¯)2Show that the expected valueof.s2is[(n−1)/n]σ2Hints: Write

(xi−x¯)2=[(xi−μ)−(x¯−μ)]2=(xi−μ)2−2(xi−μ)(x¯−μ)+(x¯−μ)2

Find the average value of the first term from the definition ofσ2and the average value of the third term from Problem 2. To find the average value of the middle term write

(x¯−μ)=(x1+x2+⋯+xnn−μ)=1n[(x1−μ)+(x2−μ)+⋯+(xn−μ)]

Show by Problemthat

E[(xi−μ)(xj−μ)]=E(xi−μ)E(xj−μ)=0   fori≠j

andevaluate6.14 (same as the first term). Collect terms to find

E(s2)=n−1nσ2

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