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Chapter 4: Q.4.6 (page 170)

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

Find c鈮�1such that EcX=1.

Short Answer

Expert verified

The two possible values of c are p and 1p-1.

Step by step solution

01

Given information

Given in the question that,

P{X=1}=p=1-P{X=-1}.

02

Calculation

Observe that random variable Xassumes only two values with positive probability. We have that

P(X=1)=p

P(X=-1)=1-p

Using the theorem about the expectation of a function of a random variable,

We have that

1=EcX=c脳p+1c(1-p)

Multiply both sides with c, we end up with an equation

pc2-c+(1-p)=0.

03

Solution

Two solutions are,

c1,2=1卤1-4p(1-p)2p

Observe that the term under the root is equal to (2p-1)2, so we have

c1,2=1卤|2p-1|2p

Therefore, we get that we have two answers

c1=1,c2=1-pp.

04

Final answer

There exist two possible values of c. They are p and 1p-1.

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Most popular questions from this chapter

In the game of Two-Finger Morra, 2players show 1or 2fingers and simultaneously guess the number of fingers their opponent will show. If only one of the players guesses correctly, he wins an amount (in dollars) equal to the sum of the fingers shown by him and his opponent. If both players guess correctly or if neither guesses correctly, then no money is exchanged. Consider a specified player, and denote by X the amount of money he wins in a single game of Two-Finger Morra.

(a) If each player acts independently of the other, and if each player makes his choice of the number of fingers he will hold up and the number he will guess that his opponent will hold up in such a way that each of the 4possibilities is equally likely, what are the possible values of Xand what are their associated probabilities?

(b) Suppose that each player acts independently of the other. If each player decides to hold up the same number of fingers that he guesses his opponent will hold up, and if each player is equally likely to hold up 1or 2 fingers, what are the possible values ofX and their associated probabilities?

There are ktypes of coupons. Independently of the types of previously collected coupons, each new coupon collected is of typeiwith probability pi, 鈭�i=1kpi=1. If n coupons are collected, find the expected number of distinct types that appear in this set. (That is, find the expected number of types of coupons that appear at least once in the set of ncoupons.)

In some military courts, 9judges are appointed. However, both the prosecution and the defense attorneys are entitled to a peremptory challenge of any judge, in which case that judge is removed from the case and is not replaced. A defendant is declared guilty if the majority of judges cast votes of guilty, and he or she is declared innocent otherwise. Suppose that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with probability .7,whereas when the defendant is, in fact, innocent, this probability drops to .3.

(a) What is the probability that a guilty defendant is declared guilty when there are (i) 9, (ii) 8, and (iii) 7judges?

(b) Repeat part (a) for an innocent defendant.

(c) If the prosecuting attorney does not exercise the right to a peremptory challenge of a judge, and if the defense is limited to at most two such challenges, how many challenges should the defense attorney make if he or she is 60percent certain that the client is guilty?

A total of 2npeople, consisting of nmarried couples, are randomly divided into npairs. Arbitrarily number the women, and let Widenote the event that woman iis paired with her husband.

  1. FindP(Wi).
  2. For i鈮�j,find role="math" localid="1646662043709" PWi鈭�Wj.
  3. When nis large, approximate the probability that no wife is paired with her husband.
  4. If each pairing must consist of a man and a woman, what does the problem reduce to?

Let N be a nonnegative integer-valued random variable. For nonnegative values aj, j 脷 1, show that

鈭�j=1鈭�a1+鈥�+ajP{N=j}=鈭�i=1鈭�aiP[N鈮�i}

Then show that

E[N]=鈭�i=1鈭�P[N鈮�i}

and

E[N(N+1)]=2鈭�i=1鈭�iP{N鈮�i}
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