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Chapter 6: Q.6.10 (page 275)

The joint probability density function of X and Y is given by f(x, y) = e-(x+y) 0 鈥� x < q, 0 鈥� y < q Find

(a) P{X < Y} and

(b) P{X < a}.

Short Answer

Expert verified

a. P{X<Y}=12

b.P{X<a}=1-e-a

Step by step solution

01

Content Introduction

The likelihood of an occasion is the extent of times the occasion occurs out of an enormous number of preliminaries.

02

Explanation (Part a)

Observe that their joint density function can be factorized as:

f(x,y)=e-(x+y)=e-xe-y=fX(x)fY(y)

So because X and Y are equally distributed random variables with distribution Expo(1) and they are independent since the joint density function can be factorized. Because of the equal distribution, we have that

P(X<Y)=12

03

Explanation (part b)

Use the fact that X~Expo(1)to obtain that

P(X<a)=P(X鈮�a)=FX(a)=1-e-afora鈮�0, otherwise it is equal to zero.

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Most popular questions from this chapter

In Problem 6.2, suppose that the white balls are numbered, and let Yiequal 1if the ith white ball is selected and0 otherwise. Find the joint probability mass function of

(a)Y1,Y2

(b)Y_{1},Y_{2},Y_{3}

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.

Show that f(x, y) = 1/x, 0 < y < x < 1, is a joint density function. Assuming that f is the joint density function of X, Y, find

(a) the marginal density of Y;

(b) the marginal density of X;

(c) E[X]; (d) E[Y].

6. Let X and Y be continuous random variables with joint density function f(x,y)=x5+cy鈥呪赌呪赌呪赌�0<x<1,1<y<50鈥呪赌呪赌呪赌�otherwise

where c is a constant.

(a) What is the value of c?

(b) Are X and Y independent?

(c) FindP[X+Y>3]

Let X1,...,Xnbe a set of independent and identically distributed continuous random variables having distribution function F, and let X(i),i=1,...,ndenote their ordered values. If X, independent of theXi,i=1,...,n, also has distribution F, determine

(a) P{X>X(n)};

(b) P{X>X(1)};

(c) PX{(i)<X<X(j)},1鈥�i<j鈥�n.
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