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)

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

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

Explanation (Part d)

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

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

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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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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

Theoretical and Mathematical Physics

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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}