Q.6.19 Let X1, X2, X3 be independent an... [FREE SOLUTION]

91影视

A First Course in Probability

Sheldon M. Ross

$Math Studyset 91影视 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 6: Q.6.19 (page 276)

Let X1, X2, X3 be independent and identically distributed continuous random variables. Compute
(a) P{X1 > X2|X1 > X3};
(b) P{X1 > X2|X1 < X3};
(c) P{X1 > X2|X2 > X3};
(d) P{X1 > X2|X2 < X3}

Short Answer

Expert verified

a. $P {X_{1} > X_{2} | X_{1} < X_{3}} = \frac{2}{3}$

b. $P {X_{1} > X_{2} | X_{1} < X_{3}} = \frac{1}{3}$

c. $P {X_{1} > X_{2} | X_{2} > X_{3}} = \frac{1}{3}$

d. $P {X_{1} > X_{2} | X_{2} < X_{3}} = \frac{2}{3}$

Step by step solution

Content Introduction

If all Xi are mutually independent and have (or belong to) the same distribution, we say that random variables $X_{1}, X_{2}, . . ., X_{n}$ are all independent and identically distributed.

Explanation (Part a)

We have that,

$P {X_{1} > X_{2} | X_{1} < X_{3}} = \frac{P (X_{1} > X_{2}, X_{1} > X_{3})}{P (X_{1} > X_{2})} = \frac{P (X_{3} < X_{2} < X_{1}) + P (X_{2} < X_{3} < X_{1})}{P (X_{1} > X_{2})} = \frac{2 . \frac{1}{6}}{\frac{1}{2}} = \frac{2}{3}$

Explanation (Part b)

$P {X_{1} > X_{2} | X_{1} < X_{3}} = \frac{P (X_{1} > X_{2}, X_{1} < X_{3})}{P (X_{1} < X_{3})} = \frac{P (X_{2} < X_{1} < X_{3})}{P (X_{1} < X_{3})} = \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{3}$

Explanation (Part c)

$P {X_{1} > X_{2} | X_{2} > X_{3}} = \frac{P (X_{1} > X_{2}, X_{2} > X_{3})}{P (X_{2} > X_{3})}$

$= \frac{P (X_{3} < 鈥� X_{2} < X_{1})}{P (X_{2} > X_{3})} = \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{3}$

Explanation (Part d)

$P {X_{1} > X_{2} | X_{2} < X_{3}} = \frac{P (X_{1} > X_{2}, X_{2} < X_{3})}{P (X_{2} < X_{3})}$

$= \frac{P (X_{2} < X_{3} < X_{1}) + P (X_{2} < X_{1} < X_{3})}{P (X_{2} < 鈥� X_{3})} = \frac{2}{3}$

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Let X1, X2, X3 be independent and identically distributed continuous random variables. Compute
(a) P{X1 > X2|X1 > X3};
(b) P{X1 > X2|X1 < X3};
(c) P{X1 > X2|X2 > X3};
(d) P{X1 > X2|X2 < X3}

Short Answer

Step by step solution

Content Introduction

Explanation (Part a)

Explanation (Part b)

Explanation (Part c)

Explanation (Part d)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Applied Mathematics

Pure Maths

Logic and Functions

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

91影视

Let X1, X2, X3 be independent and identically distributed continuous random variables. Compute(a) P{X1 > X2|X1 > X3};(b) P{X1 > X2|X1 < X3};(c) P{X1 > X2|X2 > X3};(d) P{X1 > X2|X2 < X3}

Short Answer

Step by step solution

Content Introduction

Explanation (Part a)

Explanation (Part b)

Explanation (Part c)

Explanation (Part d)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Applied Mathematics

Pure Maths

Logic and Functions

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Let X1, X2, X3 be independent and identically distributed continuous random variables. Compute
(a) P{X1 > X2|X1 > X3};
(b) P{X1 > X2|X1 < X3};
(c) P{X1 > X2|X2 > X3};
(d) P{X1 > X2|X2 < X3}