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A First Course in Probability

Sheldon M. Ross

$Math Studyset 魅影直播 Explanations$ Math

9th Edition 路 432 pages

ISBN: 9780321794772

Chapter 6: Q.6.10 (page 271)

$T h e j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n o f X a n d Y i s g i v e n b y f (x, y) = e 鈭� (x + y) 0 鈥� x < q, 0 鈥� y < q F i n d (a) P X < Y a n d (b) P X < a$

Short Answer

Expert verified

(a) $P (X < Y) = \frac{1}{2}$

$(b) P (X < a) = 1 - e^{- a}$

Step by step solution

01

Introduction

The joint probability density function of X and Y $f (x, y) = e 鈭� (x + y) 0 鈥� x < q, 0 鈥� y < q$

02

Explanation of (a).

G i v e n i n f o r m a t i o n : T h e j o i n t p r o b a b i l i t y d e n s i t y f u n c t i o n i s f (x, y) = e^{- (x + y)} 鈮� x 鈮� 鈭�, 0 鈮� y 鈮� 鈭� formula used: P (X < Y) = {鈭�}_{0}^{鈭�} {鈭�}_{0}^{y} f (x, y) d x d y {鈭�}_{0}^{鈭�} {鈭�}_{0}^{y} e^{- x} 脳 e^{- y} d x d y {鈭�}_{0}^{鈭�} {[- e^{- x}]}_{0}^{y} 脳 e^{- y} d y {鈭�}_{0}^{鈭�} (- e^{- y} + 1) 脳 e^{- y} d y {鈭�}_{0}^{鈭�} (e^{- y} - e^{- 2 y}) d y S o l v i n g i t f u r t h e r {[- e^{- y} - \frac{e^{- 2 y}}{- 2}]|}_{0}^{鈭�} = 1 - \frac{1}{2} = \frac{1}{2}

03

Explanation of (b).

Formula used:

P (X < a) = {鈭�}_{0}^{鈭�} {鈭�}_{0}^{a} f (x, y) d x d y C a l c u l a t i o n : F i n d t h e r e q u i r e d p r o b a b i l i t y u \sin g t h e a b o v e f o r m u l a {鈭�}_{0}^{鈭�} {鈭�}_{0}^{a} e^{- x} 脳 e^{- y} d x d y {鈭�}_{0}^{鈭�} {[- e^{- x}]}_{0}^{a} 脳 e^{- y} d y {鈭�}_{0}^{鈭�} (- e^{- a} + 1) 脳 e^{- y} d y (- e^{- a} + 1) 脳 {\frac{e^{- y}}{- 1}|}_{0}^{鈭�} = 1 - e^{- a}

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Most popular questions from this chapter

Verify Equation $(1.2)$ . $P (a_{1} < X 鈮� a_{2}, b_{1} < Y 鈮� b_{2}) = F (a_{2}, b_{2}) + F (a_{1}, b_{1}) 鈭� F (a_{1}, b_{2}) 鈭� F (a_{2}, b_{1})$

Let $X 1, X 2, . . .$ be a sequence of independent uniform $(0, 1)$ random variables. For a fixed constant c, define the random variable N by $N = m i n {n : X n > c}$ Is N independent of $X_{N}$ ? That is, does knowing the value of the first random variable that is greater than c affect the probability distribution of when this random variable occurs? Give an intuitive explanation for your answer.

The joint density of X and Y is

$f (x, y) = c (x^{2} - y^{2}) e^{- x} 0 鈮� x < 鈭�, - x 鈮� y 鈮� x$

Find the conditional distribution of Y, given X = x.

A bin of 5 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by N₁ the number of tests made until the first defective is identified and by N₂ the number of additional tests until the second defective is identified. Find the joint probability mass function of N₁ and N₂.

Consider an urn containing n balls numbered $1, . . . . ., n$ and suppose that k of them are randomly withdrawn. Let $X_{i}$ equal $1$ if ball number $i$ is removed and let $X_{i}$ be $0$ otherwise. Show that $X_{1}, . . . . . . X_{n}$ are exchangeable .

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