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Chapter 4: Q. 4.19 (page 164)

If the distribution function of Xis given by

F(b)=0鈥呪赌呪赌呪赌�b<012鈥呪赌呪赌呪赌�0鈮�b<135鈥呪赌呪赌呪赌�1鈮�b<245鈥呪赌呪赌呪赌�2鈮�b<3910鈥呪赌呪赌呪赌�3鈮�b<3.51鈥呪赌呪赌呪赌�b鈮�3.5

calculate the probability mass function of X.

Short Answer

Expert verified

p(x)=12鈥呪赌呪赌呪赌�x=0110鈥呪赌呪赌呪赌�x=115鈥呪赌呪赌呪赌�x=2110鈥呪赌呪赌呪赌�x=3110鈥呪赌呪赌呪赌�x=3.5

Step by step solution

01

Given information

F(b)=0鈥呪赌呪赌呪赌�b<012鈥呪赌呪赌呪赌�0鈮�b<135鈥呪赌呪赌呪赌�1鈮�b<245鈥呪赌呪赌呪赌�2鈮�b<3910鈥呪赌呪赌呪赌�3鈮�b<3.51鈥呪赌呪赌呪赌�b鈮�3.5

02

Explanation

The probability mass procedure is equivalent to zero for every particular where the distribution function is locally equivalent to some consistent. Therefore, we only have to think of facts in which function Fhas a brake.

Here,

P(0)=F(0)-P(b<0)

=12

P(1)=35-12

=110

P(2)=45-35

=15

P(3)=910-45

=110

P(3.5)=1-910

=110

03

Final answer

Probability mass function is

p(x)=12鈥呪赌呪赌呪赌�x=0110鈥呪赌呪赌呪赌�x=115鈥呪赌呪赌呪赌�x=2110鈥呪赌呪赌呪赌�x=3110鈥呪赌呪赌呪赌�x=3.5

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Most popular questions from this chapter

Let Xbe such thatP{X=1}=p=1-P{X=-1}

Find c鈮�1such that EcX=1.

Each night different meteorologists give us the probability that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability p, then he or she will receive a score of

1鈭�(1鈭�p)2if it does rain

1-p2ifitdoesnotrain

We will then keep track of scores over a certain time span and conclude that the meteorologist with the highest average score is the best predictor of weather. Suppose now that a given meteorologist is aware of our scoring mechanism and wants to maximize his or her expected score. If this person truly believes that it will rain tomorrow with probability p*, what value of p should he or she assert so as to maximize the expected score?

Five distinct numbers are randomly distributed to players numbered1through 5. Whenever two players compare their numbers, the one with the higher one is declared the winner. Initially, players1and 2 compare their numbers; the winner then compares her number with that of player 3, and so on. Let X denote the number of times player 1 is a winner. FindPX=i,i=0,1,2,3,4.

Suppose that the distribution function of X given by

F(b)=0鈥呪赌呪赌呪赌�b<0b4鈥呪赌呪赌呪赌�0鈮�b<112+b鈭�14鈥呪赌呪赌呪赌�1鈮�b<21112鈥呪赌呪赌呪赌�2鈮�b<31鈥呪赌呪赌呪赌�3鈮�b

(a) Find P{X=i},i=1,2,3.

(b) Find P12<X<32.

Find Var(X) and Var(Y) for X and Y as given in Problem 4.21
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