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Chapter 4: Q.4.15 (page 174)

The number of eggs laid on a tree leaf by an insect of a certain type is a Poisson random variable with parameter 位. However, such a random variable can be observed only if it is positive, since if it is 0, then we cannot know that such an insect was on the leaf. If we let Ydenote the observed number of eggs, then

P{Y=i}=P{X=i鈭�X>0}

where Xis Poisson with parameter 位. Find E[Y].

Short Answer

Expert verified

The value ofEY=位1-e-位

Step by step solution

01

Given information

Given in the question that , The number of eggs laid on a tree leaf by an insect of a certain type is a Poisson random variable with parameter 位. However, such a random variable can be observed only if it is positive, since if it is 0 , then we cannot know that such an insect was on the leaf. If we let Ydenote the observed number of eggs, then

P{Y=i}=P{X=i鈭�X>0}

where Xis Poisson with parameter 位.

02

Explanation

We have,

P(Y=i)=P(X=i鈭�X>0)=P(X=i)P(X>0)=P(X=i)1-P(X=0)

=11-e-位P(X=i)

Using the definition of expectation, we have that

EY=鈭�i=1鈭�iP(Y=i)=11-e-位鈭�i=1鈭�iP(X=i)

=11-e-位鈭�i=0鈭�iP(X=i)=11-e-位EX

=位1-e-位

03

Step 3:Final answer

EY=位1-e-位

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Most popular questions from this chapter

Let Xbe a negative binomial random variable with parameters rand p, and let Ybe a binomial random variable with parameters nand p. Show that

P{X>n}=P{Y<r}

Hint: Either one could attempt an analytical proof of the preceding equation, which is equivalent to proving the identity

鈭�i=n+1鈭�鈥�i鈭�1r鈭�1pr(1鈭�p)i鈭�r=鈭�i=0r鈭�1鈥�ni脳pi(1鈭�p)ni

or one could attempt a proof that uses the probabilistic interpretation of these random variables. That is, in the latter case, start by considering a sequence of independent trials having a common probability p of success. Then try to express the events to express the events {X>n}and {Y<r}in terms of the outcomes of this sequence.

Four buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, 40, 33, 25, and 50 students. One of the students is randomly selected. Let X denote the number of students who were on the bus carrying the randomly selected student. One of the 4 bus drivers is also randomly selected. Let Y denote the number of students on her bus.

(a) Which of E[X] or E[Y] do you think is larger? Why?

(b) Compute E[X] and E[Y].

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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.
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