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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

The joint density of X and Y is

f(x,y)=cx2-y2e-x0鈮�x<鈭�,-x鈮�y鈮�x

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

The joint density function of X and Y is given by

f(x,y)=xe-x(y+1)x>0,y>0

(a) Find the conditional density of X, given Y = y, and that of Y, given X = x.

(b) Find the density function of Z = XY.

Let X and Y be independent uniform (0, 1) random variables.

(a) Find the joint density of U = X, V = X + Y.

(b) Use the result obtained in part (a) to compute the density function of V

If X and Y are independent binomial random variables with identical parameters n and p, show analytically that the conditional distribution of X given that X + Y = m is the hypergeometric distribution. Also, give a second argument that yields the same result without any computations. Hint: Suppose that 2n coins are flipped. Let X denote the number of heads in the first n flips and Y the number in the second n flips. Argue that given a total of m heads, the number of heads in the first n flips has the same distribution as the number of white balls selected when a sample of size m is chosen from n white and n black balls

If U is uniform on (0,2蟺)and Z, independent of U, is exponential with rate 1, show directly (without using the results of Example 7b) that X and Y defined by

X=2ZcosUY=2ZsinU

are independent standard normal random variables.
See all solutions

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