Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Q. 8.17 (page 391)

Redo Example5bunder the assumption that the number of man-woman pairs is (approximately) normally distributed. Does this seem like a reasonable supposition?

Short Answer

Expert verified

Therefore,

P{X≤30}≈0.

Step by step solution

01

Given Information.

the number of man-woman pairs is (approximately) normally distributed.

02

Explanation.

Consider Example5b(textbook). Now, assume that the number of man-woman pairs

X=∑i=1100Xi,whereXi=1,maniis paired with a woman0,otherwise

is approximately normally distributed with a meanE[X]≈50.25and a variance Var(X)≈25.126

The probability that most 30pairs will consist of a man and a woman isP{X≤30}. To approximate this probability we can use the central limit theorem and in that case, we get:

localid="1649832005870" P{X≤30}≈PX-E[X]Var(X)≤30-E[X]Var(X)localid="1649831889184" =PX-50.2525.126≤30-50.2525.126

≈0.

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

LetZn, n≥1be a sequence of random variables andca constant such that for each

ε>0,P|Zn−c|>ε→0asn→q. Show that for any bounded continuous functiong,

E[g(Zn)]→g(c)asn→q.

In Problem8.7, suppose that it takes a random time, uniformly distributed over(0,.5), to replace a failed bulb. Approximate the probability that all bulbs have failed by time550.

A certain component is critical to the operation of an electrical system and must be replaced immediately upon failure. If the mean lifetime of this type of component is 100 hours and its standard deviation is 30 hours, how many of these components must be in stock so that the probability that the system is in continual operation for the next 2000 hours is at least .95?

A.J. has 20 jobs that she must do in sequence, with the times required to do each of these jobs being independent random variables with mean 50 minutes and standard deviation 10 minutes. M.J. has 20 jobs that he must do in sequence, with the times required to do each of these jobs

being independent random variables with mean 52 minutes and standard deviation 15 minutes.

(a) Find the probability that A.J. finishes in less than 900 minutes.

(b) Find the probability that M.J. finishes in less than 900 minutes.

(c) Find the probability that A.J. finishes before M.J.

Let f(x)be a continuous function defined for0≤x≤1. Consider the functions

Bn(x)=∑k=0nfknnkxk(1−x)n-k(called Bernstein polynomials) and prove that

limn→∞Bn(x)=f(x).

Hint: Let X1,X2,...be independent Bernoulli random variables with meanx. Show that

Bn(x)=Efx1+...+xnn

and then use Theoretical Exercise8.4.

Since it can be shown that the convergence of Bn(x)to f(x)is uniformx, the preceding reasoning provides a probabilistic proof of the famous Weierstrass theorem of analysis, which states that any continuous function on a closed interval can be approximated arbitrarily closely by a polynomial.
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Geometry

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