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Chapter 6: Q.6.22 (page 272)

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].

Short Answer

Expert verified

a. X and Y are not independent.

b. The density function of X is

fX(x)=x+12;0<x<1

c. The value of P[X+Y<1]=23

Step by step solution

01

Content Introduction

X and Y are random variables with a combined density function.

02

Explanation (Part a)

The joint density function of X and Y is

fX,Y(x,y)=x+y;0<x<1and0<y<1

FormarginaldensityofX,pleaserefferpart(b)forexplanationfX(x)=x+0.5;0<x<1Thereforebysymmetry,forY,wehavefY(y)=y+0.5;0<y<1(x+12)脳(y+12)鈮�x+yWhichprovesthatxandyaredependent.

03

Explanation (Part b)

The marginal density of X is

fX(x)=鈭�01(x+y)dy=xy+y2201=x+12

x+120<x<1

04

Explanation (Part c)

P[X+Y<1]=鈭�x=01鈭�y=01-xfX,Y(x,y)dydx=鈭�01鈭�01-x(x+y)dy=鈭�01xy+y2201-xdxUponsimplification,weobtain=鈭�01x2+12dx=x36+x201=23

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Most popular questions from this chapter

If X and Y are independent random variables both uniformly distributed over (0,1), find the joint density function of R=X2+Y2,胃=tan-1Y/X.

The joint probability density function of X and Y is given by

f(x.y)=67(x2+xy2)0<x<1,0<y<2

(a) Verify that this is indeed a joint density function.

(b) Compute the density function of X.

(c) Find P{X > Y}.

(d) Find P{Y > 1 2 |X < 1 2 }.

(e) Find E[X].

(f) Find E[Y].

Suppose that the number of events occurring in a given time period is a Poisson random variable with parameter 位. If each event is classified as a type ievent with probabilitypi,i=1,....,n,鈭�pi=1, independently of other events, show that the numbers of type ievents that occur,i=1,....,n, are independent Poisson random variables with respective parameters位pi,i=1,.....,n.

Suppose that X, Y, and Z are independent random variables that are each equally likely to be either 1 or 2. Find the probability mass function of

(a) XYZ,

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Let Z1and Z2be independent standard normal random variables. Show that X, Y has a bivariate normal distribution when X=Z1,Y=Z1+Z2.
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