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Chapter 6: Q. 6.42 (page 274)

The joint density of X and Y is

f(x,y)=cx2-y2e-x0≤x<∞,-x≤y≤x

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

Short Answer

Expert verified

For t≤x:

P(Y≤t∣X=x)=32x3x2(t+x)-t33+x33

For t<-x:

P(Y≤t∣X=x)=0

For t>x:

P(Y≤t∣X=x)=1

Step by step solution

01

Given information

Joint density of X and Y

f(x,y)=cx2-y2e-x0≤x<∞,-x≤y≤x

also X=x

Formula to be used :

f(y∣x)=f(x,y)f(x)

02

Explanation

∫f(x,y)dy=∫-xxcx2-y2e-xdx=2c3x3e-x,0<x<∞Thus,f(y∣x)=32x3x2-y2,-x≤y≤x

By integration,

For t≤x,

localid="1647240753725" role="math" P(Y≤t∣X=x)=∫-xtf(x,y)dy=32x3x2(t+x)-t33+x33

Then for t < -x,

P(Y≤t∣X=x)=0

and for t > x,

P(Y≤t∣X=x)=1

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Most popular questions from this chapter

According to the U.S. National Center for Health Statistics, 25.2 percent of males and 23.6 percent of females never eat breakfast. Suppose that random samples of 200 men and 200 women are chosen. Approximate the probability that

(a) at least 110 of these 400 people never eat breakfast;

(b) the number of the women who never eat breakfast is at least as large as the number of the men who never eat breakfast.

Let Z1,Z2......Znbe independent standard normal random variables, and let Sj=∑i=1jZi

(a) What is the conditional distribution of Sn given that Sk=y,for k = 1, ... , n?

(b) Show that, for 1 … k … n, the conditional distribution of SKgiven that

Sn = x is normal with mean xk/n and variance k(n − k)/n.

The joint density of X and Y is given by

f(x,y)=C(y-x)e-y-y<x<y,0<y<∞

(a) Find C.

(b) Find the density function of X.

(c) Find the density function of Y.

(d) Find E[X].

(e) Find E[Y].

A model proposed for NBA basketball supposes that when two teams with roughly the same record play each other, the number of points scored in a quarter by the home team minus the number scored by the visiting team is approximately a normal random variable with mean 1.5 and variance 6. In addition, the model supposes that the point differentials for the four quarters are independent. Assume that this model is correct.

(a) What is the probability that the home team wins?

(b) What is the conditional probability that the home team wins, given that it is behind by 5 points at halftime?

(c) What is the conditional probability that the home team wins, given that it is ahead by 5 points at the end of the first quarter?

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
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