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Chapter 4: Q. 4.27 (page 172)

If Xis a geometric random variable, show analytically that

P{X=n+k鈭�X>n}=P{X=k}

Using the interpretation of a geometric random variable, give a verbal argument as to why the preceding equation is true.

Short Answer

Expert verified

We proved that

P{X=n+k鈭�X>n}=P{X=k}

Step by step solution

01

Definition

A random variable that takes the value k, a non-negative integer with probability pk(1-p).

02

Calculation

P{X=n+k鈭�X>n}=P{X=n+k鈭�X>n}P(X>n)

P{X=n+k鈭�X>n}=(1-p)n+k-1p

P(X>n)=First nare failures

=(1-p)n

鈬�P{X=n+k鈭�X>n}=(1-p)n+k-1(1-p)n.p

=(1-p)k-1p

=P{X=k}

Hence proved

03

continue Calculation

Also because trials are independent

Given that first ntrials does'nt result in success

Next kthwill result in success and next (k-1)in failures =(1-p)k-1p

=P{X=k}

鈬�P{X=n+k鈭�X>n}=P{X=k}

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Most popular questions from this chapter

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.

The random variable X is said to have the Yule-Simons distribution if

P{X=n}=4n(n+1)(n+2),n鈮�1

(a) Show that the preceding is actually a probability mass function. That is, show that鈭�n=1鈭�P{X=n}=1

(b) Show that E[X] = 2.

(c) Show that E[X2] = q

From a set of nrandomly chosen people, let Eijdenote the event that persons iand jhave the same birthday. Assume that each person is equally likely to have any of the 365 days of the year as his or her birthday. Find

(a) PE3,4鈭�E1,2;

(b) PE1,3鈭�E1,2;

(c) PE2,3鈭�E1,2鈭�E1,3.

What can you conclude from your answers to parts (a)-(c) about the independence of the n2events Eij?

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

A box contains 5red and 5blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win \(1.10; if they are different colors, then you win -\)1.00. (That is, you lose $1.00.) Calculate

(a) the expected value of the amount you win;

(b) the variance of the amount you win.
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