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)

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

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

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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)

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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

Calculus

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Discrete Mathematics

Decision Maths

Pure Maths

Theoretical and Mathematical Physics

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}