Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 7: Q.7.31 (page 366)

For random variables X and Y, show that

Var⁡(X+Y)≤Var⁡(X)+Var⁡(Y)

That is, show that the standard deviation of a sum is always less than or equal to the sum of the standard deviations.

Short Answer

Expert verified

The standard deviation is the square root of variance that isVar⁡(X+Y)≤Var⁡(X)+Var⁡(Y)

Step by step solution

01

Given information

X and Y are random variable

That has the equationVar⁡(X+Y)≤Var⁡(X)+Var⁡(Y)

02

Solution 

We need to define the following equations first,

Var⁡(X)=σx2

Var⁡(Y)=σy2

The relation of variance and covariance show that,

Var⁡(X+Y)=σx2+σy2+2Cov⁡(X,Y)

Var⁡(X+Y)=σx2+σy2+2σxσy

The property of correlation Shows that,

Corr⁡(X,Y)=Cov⁡(X,Y)σxσy

The preceding inequality will becomes,

Corr⁡(X,Y)=Cov⁡(X,Y)σxσy≤1

So,

Cov⁡(X,Y)σxσy≤1

Cov⁡(X,Y)≤σxσy

03

Solution

Now use the above results to obtain the result,

Var⁡(X+Y)=σx2+σy2+2Cov⁡(X,Y)

Var⁡(X+Y)≤σx2+σy2+2σxσy

Var⁡(X+Y)≤σx+σy2

Var⁡(X+Y)≤(Var⁡(X)+Var⁡(Y))2

04

Final answer

The standard deviation is the square root of variance that isVar⁡(X+Y)≤Var⁡(X)+Var⁡(Y)

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

Consider a gambler who, at each gamble, either wins or loses her bet with respective probabilities pand 1-p. A popular gambling system known as the Kelley strategy is to always bet the fraction 2p-1of your current fortune when p>12. Compute the expected fortune afterngambles of a gambler who starts with xunits and employs the Kelley strategy.

Ten hunters are waiting for ducks to fly by. When a flock of ducks flies overhead, the hunters fire at the same time, but each chooses his target at random, independently of the others. If each hunter independently hits his target with probability .6, compute the expected number of ducks that are hit. Assume that the number of ducks in a flock is a Poisson random variable with mean 6.

In Problem 7.6, calculate the variance of the sum of the rolls.

A deck of n cards numbered 1 through n is thoroughly shuffled so that all possible n! orderings can be assumed to be equally likely. Suppose you are to make n guesses sequentially, where the ith one is a guess of the card in position i. Let N denote the number of correct guesses.

(a) If you are not given any information about your earlier guesses, show that for any strategy, E[N]=1.

(b) Suppose that after each guess you are shown the card that was in the position in question. What do you think is the best strategy? Show that under this strategy

E[N]=1n+1n−1+⋯+1≈∫1n1xdx=logn

(c) Supposethatyouaretoldaftereachguesswhetheryou are right or wrong. In this case, it can be shown that the strategy that maximizes E[N] is one that keeps on guessing the same card until you are told you are correct and then changes to a new card. For this strategy, show that

E[N]=1+12!+13!+⋯+1n!≈e−1

Hint: For all parts, express N as the sum of indicator (that is, Bernoulli) random variables.

Consider nindependent trials, the ithof which results in a success with probability Pl.

(a) Compute the expected number of successes in the ntrials-call it μ

(b) For a fixed value of μ, what choice of P1,…,Pnmaximizes the variance of the number of successes?

(c) What choice minimizes the variance?
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.

Sign-up for free