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Chapter 7: Q.57 (page 356)

Suppose that the expected number of accidents per week at an industrial plant is 5. Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of 2.5. If the number of workers injured in each accident is independent of the number of accidents that occur, compute the expected number of workers injured in a week .

Short Answer

Expert verified

E(NX)=12.5

Step by step solution

01

Given information

Given in the question that, .Suppose that the expected number of accidents per week at an industrial plant is 5. Suppose also that the numbers of workers injured in each accident are independent random variables with a common mean of 2.5.

02

Explanation

Let N= number of accidents to happen in seven days. Allow X=to number of workers injured in every mishap. The quantity of workers injured in accidents each week should be N脳X. So we needs E[NX]

Since we know that Nand Xare independent, we know that

Cov(N,X)=E(NX)-E(N)E(X)=0

鈬�E(NX)=E(N)E(X)=5脳2.5=12.5

03

Final answer

The expected number of workers injured in a week isE(NX)=12.5

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Most popular questions from this chapter

Two envelopes, each containing a check, are placed in front of you. You are to choose one of the envelopes, open it, and see the amount of the check. At this point, either you can accept that amount or you can exchange it for the check in the unopened envelope. What should you do? Is it possible to devise a strategy that does better than just accepting the first envelope? Let Aand B, A<B, denote the (unknown) amounts of the checks and note that the strategy that randomly selects an envelope and always accepts its check has an expected return of (A+B)/2. Consider the following strategy: Let F(路)be any strictly increasing (that is, continuous) distribution function. Choose an envelope randomly and open it. If the discovered check has the value x, then accept it with probability F(x)and exchange it with probability 1鈭�F(x).

(a) Show that if you employ the latter strategy, then your expected return is greater than (A+B)/2. Hint: Condition on whether the first envelope has the value Aor B. Now consider the strategy that fixes a value x and then accepts the first check if its value is greater than x and exchanges it otherwise. (b) Show that for any x, the expected return under thex-strategy is always at least (A+B)/2and that it is strictly larger than (A+B)/2if xlies between Aand B.

(c) Let X be a continuous random variable on the whole line, and consider the following strategy: Generate the value ofX, and if X=x, then employ the x-strategy of part (b). Show that the expected return under this strategy is greater than (A+B)/2.

The random variables X and Y have a joint density function is given by

f(x,y)={2e鈭�2x/x0鈮�x<鈭�,0鈮�y鈮�x0otherwise

ComputeCov(X,Y)

Let A1,A2,鈥�,Anbe arbitrary events, and define

Ck={at least kof the Aioccur}. Show that

鈭�k=1nPCk=鈭�k=1nPAk

Hint: Let Xdenote the number of the Aithat occur. Show

that both sides of the preceding equation are equal to E[X].

A group of nmen and n women is lined up at random.

(a) Find the expected number of men who have a woman next to them.

(b) Repeat part (a), but now assuming that the group is randomly seated at a round table.

Suppose that A and B each randomly and independently choose3of10objects. Find the expected number of objects

a. Chosen by both A and B;

b. Not chosen by either A or B;

c. Chosen by exactly one of A and B.

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