/*! 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. 4.27 If X聽is a geometric random vari... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Q. 4.27 (page 172)

If Xis a geometric random variable, show analytically that

P{X=n+k鈭�X>n}=P{X=k}

Using the interpretation of a geometric random variable, give a verbal argument as to why the preceding equation is true.

Short Answer

Expert verified

We proved that

P{X=n+k鈭�X>n}=P{X=k}

Step by step solution

01

Definition

A random variable that takes the value k, a non-negative integer with probability pk(1-p).

02

Calculation

P{X=n+k鈭�X>n}=P{X=n+k鈭�X>n}P(X>n)

P{X=n+k鈭�X>n}=(1-p)n+k-1p

P(X>n)=First nare failures

=(1-p)n

鈬�P{X=n+k鈭�X>n}=(1-p)n+k-1(1-p)n.p

=(1-p)k-1p

=P{X=k}

Hence proved

03

continue Calculation

Also because trials are independent

Given that first ntrials does'nt result in success

Next kthwill result in success and next (k-1)in failures =(1-p)k-1p

=P{X=k}

鈬�P{X=n+k鈭�X>n}=P{X=k}

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

Consider Problem 4.22 with i = 2. Find the variance of the number of games played, and show that this number is maximized when p = 1 2 .

The probability of being dealt a full house in a hand of poker is approximately .0014. Find an approximation for the probability that in 1000 hands of poker, you will be dealt at least 2 full houses.

If X has distribution function F, what is the distribution function of the random variable 伪X + 尾, where 伪 and 尾 are constants, 伪鈮�0?

Show how the derivation of the binomial probabilities P{X=i}=nipi(1-p)n-i,i=0,鈥�,nleads to a proof of the binomial theorem (x+y)n=鈭�i=0nnixiyn-iwhen xand yare nonnegative.

Hint: Let p=xx+y.

The suicide rate in a certain state is 1 suicide per 100,000 inhabitants per month.

(a) Find the probability that in a city of 400,000 inhabitants within this state, there will be 8 or more suicides in a given month.

(b) What is the probability that there will be at least 2 months during the year that will have 8 or more suicides?

(c) Counting the present month as month number 1, what is the probability that the first month to have 8 or more suicides will be month number i,i鈮�1? What assumptions are you making?
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Mechanics 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.