/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Q.7.43 Show that for random variables X... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Q.7.43 (page 362)

Show that for random variables X and Z.

E(X-Y)2=EX2-EY2

whereY=E[X鈭�Z]

Short Answer

Expert verified

X-E(X鈭�Z) is orthogonal to E(X鈭�Z).

Step by step solution

01

Given Information

Y=E[X鈭�Z]

02

Explanation

Write out the left side. We have that

E(X-Y)2=EX2+Y2-2XY

=EX2+EY2-2E(XY)

Now, what is E(XY)? Let's prove that it is equal to EY2.

EXY-Y2=E(Y(X-Y))

=E(E(X鈭�Z)路(X-E(X鈭�Z)))=0

03

Explanation

The expression above is equal to zero, since we know that the projection E(X鈭�Z)and the connection between the point Xand the projection E(X鈭�Z)are orthogonal. Therefore E(XY)=EY2which implies

E(X-Y)2=EX2-EY2.

04

Final Answer

X-E(X鈭�Z) is orthogonal to E(X鈭�Z).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

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鈮�x鈮�1,E[N(x)]=ex, where

N(x)=minn:鈭�i=1n鈥�Ui>x

This problem gives another approach to establishing that result.

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

P{N(x)鈮�n+1}=xnn!

Hint: First condition onU1and then use the induction hypothesis.

use part (a) to conclude that

E[N(x)]=ex

Let X1,X2,鈥�,Xnbe independent random variables having an unknown continuous distribution function Fand let Y1,Y2,鈥�,Ymbe independent random variables having an unknown continuous distribution function G. Now order those n+mvariables, and let

Ii=1鈥呪赌呪赌呪赌�if theith smallest of then+m鈥呪赌呪赌呪赌�variables is from theXsample0鈥呪赌呪赌呪赌�otherwise

The random variable R=鈭�i=1n+miIiis the sum of the ranks of the Xsample and is the basis of a standard statistical procedure (called the Wilcoxon sum-of-ranks test) for testing whether Fand Gare identical distributions. This test accepts the hypothesis that F=Gwhen Ris neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of R.

Hint: Use the results of Example 3e.

The joint density of Xand Yis given by

f(x,y)=e-yy,0<x<y,0<y<鈭�

Compute EX3鈭�Y=y.

Let Xbe the number of 1鈥�sand Ythe number of 2'sthat occur in nrolls of a fair die. Compute Cov(X,Y).

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

f(x,y)={2e鈭�2x/x0鈮�x<鈭�,0鈮�y鈮�x0otherwise

ComputeCov(X,Y)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.