Q.6.8 Let X and Y be independent conti... [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.6.8 (page 275)

Let X and Y be independent continuous random variables with respective hazard rate functions 位X(t) and 位Y(t), and set W = min(X, Y).
(a) Determine the distribution function of W in terms of those of X and Y.
(b) Show that 位W(t), the hazard rate function of W, is given by 位W(t) = 位X(t) + 位Y(t)

Short Answer

Expert verified

a. $F_{w} (t) = 1 - (1 - F_{x} (t)) (1 - F_{y} (t))$

b. $位_{w} (t) = 位_{x} (t) + 位_{y} (t)$

Step by step solution

Content Introduction

X and Y are independent continuous random variable with respect hazard rate function $位_{x} (t)$ and $位_{y} (t)$

Explanation (Part a)

Let us determine the cumulative distribution function of W using the fact that W is larger than t when both X and Y are larger than t.

$F_{w} (t) = P (W 鈮� t) = 1 - P (W > t) = 1 - P (X > t, Y > t)$

X and Y are both independent

$= 1 - P (X > t) (Y > t) = 1 - (1 - P (X 鈮� t)) (1 - P (Y 鈮� t)) = 1 - (1 - F_{x} (t)) (1 - F_{y} (t))$

Explanation (Part b)

We use the definition of the hazard rate function $位 (t) = \frac{f (t)}{1 - F (t)}$

$位_{w} (t) = \frac{f_{w} (t)}{1 - F_{w} (t)} = \frac{(1 - F_{x} (t)) f_{y} (t) + (1 - F_{y} (t)) f_{x} (t)}{1 - (1 - (1 - F_{x} (t)) (1 - F_{y} (t)} = \frac{(1 - F_{x} (t)) f_{y} (t)}{1 - (1 - (1 - F_{x} (t)) (1 - F_{y} (t)} + \frac{(1 - F_{y} (t)) f_{x} (t}{1 - (1 - (1 - F_{x} (t)) (1 - F_{y} (t)} = \frac{f_{y} (t)}{1 - F_{y} (t)} + \frac{f_{x} (t)}{1 - F_{x} (t)} = 位_{x} (t) + 位_{y} (t)$

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91影视

Short Answer

Step by step solution

Content Introduction

Explanation (Part a)

Explanation (Part b)

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