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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

The joint density function of X and Y isf(x,y)=x+y0<x<1,0<y<100otherwise

(a) Are X and Y independent?

(b) Find the density function of X.

(c) FindP[X+Y<1].

Suppose that106people arrive at a service station at times that are independent random variables, each of which is uniformly distributed over0,106. Let N denote the number that arrive in the first hour. Find an approximation forPN=i.

If X and Y are independent continuous positive random variables, express the density function of (a) Z = X/Y and (b) Z = XY in terms of the density functions of X and Y. Evaluate the density functions in the special case where X and Y are both exponential random variables

The random vector (X, Y) is said to be uniformly distributed over a region R in the plane if, for some constant c, its joint density is f(x, y) =  c if(x, y) 鈭� R 0 otherwise

(a) Show that 1/c = area of region R. Suppose that (X, Y) is uniformly distributed over the square centered at (0, 0) and with sides of length 2

(b) Show that X and Y are independent, with each being distributed uniformly over (鈭�1, 1).

(c) What is the probability that (X, Y) lies in the circle of radius 1 centered at the origin? That is, find P{X2 + Y2< 1}.

Let U denote a random variable uniformly distributed over (0, 1). Compute the conditional distribution of U given that

(a) U > a;

(b) U < a; where 0 < a < 1.
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