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Chapter 7: Q.7.19 (page 360)

Show that Xand Yare identically distributed and not necessarily independent, thenCov(X+Y,X-Y)=0.

Short Answer

Expert verified

It has been show thatCov(X+Y,X-Y)=0

Step by step solution

01

Given Information

Identically distributed and not necessarily independent variable=X,Y

ShowCov(X+Y,X-Y)=0

02

Explanation

We know that,

Cov(X+Y,X−Y)=Cov(X,X)+Cov(X,−Y)+Cov(Y,X)+Cov(Y,−Y)

=Var(X)−Cov(X,Y)+Cov(X,Y)−Var(Y)

=Var(X)−Var(Y)

Now As Xand Yare identically Distributed

⇒Var(X)=Var(Y)=σ2

And Cov(X,Y)=Cov(Y,X)=0

03

Final Answer

It has been shown thatCov(X+Y,X-Y)=0

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Most popular questions from this chapter

LetU1,U2,...be a sequence of independent uniform(0,1)random variables. In Example 5i, we showed that for 0≤x≤1,E[N(x)]=ex, where

N(x)=minn:∑i=1n Ui>x

This problem gives another approach to establishing that result.

(a) Show by induction on n that for 0<x≤10 and all n≥0

P{N(x)≥n+1}=xnn!

Hint: First condition onU1and then use the induction hypothesis.

use part (a) to conclude that

E[N(x)]=ex

The random variables X and Y have a joint density function is given by

f(x,y)={2e−2x/x0≤x<∞,0≤y≤x0otherwise

ComputeCov(X,Y)

7.2. Suppose that Xis a continuous random variable with

density function f. Show that E[IX-a∣]is minimized

when ais equal to the median of F.

Hint: Write

E[IX-al]=|x-a|f(x)dx

Now break up the integral into the regions where x<a

and where x>a, and differentiate.

Let Z be a standard normal random variable,and, for a fixed x, set

X={ZifZ>x0otherwise

Show thatE[X]=12πe−x2/2.

If X1,X2,X3, and X4are (pairwise) uncorrelated random variables, each having mean 0 and variance 1 , compute the correlations of

(a) X1+X2andX2+X3

(b) X1+X2and X3+X4.
See all solutions

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