/*! 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.6.9 Let X1, ... , Xn be independent ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Q.6.9 (page 275)

Let X1, ... , Xn be independent exponential random variables having a common parameter 位. Determine the distribution of min(X1, ... , Xn)

Short Answer

Expert verified

The minimum distribution isZ~Expo(n位)

Step by step solution

01

Content Introduction

In a Poisson point process, when events occur continuously and independently at a constant average rate, the exponential distribution is the probability distribution of the time between occurrences. It's an example of the gamma distribution in action.

02

Content Explanation

Define random variable

Z=min(X1,X2,......Xn)

Because of Xi > 0 we have that Z > 0. We have that

P(Z鈮�z)=1-P(Z>z)=1-P(X1>z,......,Xn>z)=1鈭�i=1nP((X1>z)=1-P(X1>z)n =1-e-n位z

So, we see thatZ~Expo(n位)

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

Suppose X and Y are both integer-valued random variables. Let p(i|j) = P(X = i|Y = j) and q(j|i) = P(Y = j|X = i) Show that P(X = i, Y = j) = p(i|j)  i p(i|j) q(j|i

If X1,X2,X3are independent random variables that are uniformly distributed over0,1, compute the probability that the largest of the three is greater than the sum of the other two.

Suppose that 3balls are chosen without replacement from an urn consisting of 5white and8red balls. Let role="math" localid="1649430608157" Xiequal 1if the ith ball selected is white, and let it equal 0otherwise. Give the joint probability mass function of

(a) X1,X2;

(b)X1,X2,X3 .

Let X and Y be independent uniform (0, 1) random variables.

(a) Find the joint density of U = X, V = X + Y.

(b) Use the result obtained in part (a) to compute the density function of V

Choose a number X at random from the set of numbers 1,2,3,4,5. Now choose a number at random from the subset no larger than X, that is, from 1...,X. Call this second number Y.

(a) Find the joint mass function of X and Y.

(b) Find the conditional mass function of X given that Y = i. Do it for i = 1,2,3,4,5.

(c) Are X and Y independent? Why?
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Logic and Functions

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

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.