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Chapter 7: Q.7.17 (page 364)

A total ofmitems are to be sequentially distributed among ncells, with each item independently being put in a cell jwith probability role="math" localid="1647437056272" pj,j=1,...,n. Find the expected number of collisions that occur, where a collision occurs whenever an item is put into a non-empty cell.

Short Answer

Expert verified

The expected number of collisions that occur, where a collision occurs whenever an item is put into a non-empty cell,

E[X]=m鈭�n+鈭�j=1n鈥�1鈭�pjm

Step by step solution

01

Step 1:Concept Introduction

Given that

Xi=1if a collision occurs when the item is placed

=0otherwise

X=Totalnumberofcollisions

=鈭�i=1m鈥�Xi

鈬�E[X]=鈭�i=1m鈥�EXi

02

Step 2:Explanation

Molding on the cell in which it is set.

EXi=鈭�j鈥�EXi鈭�placed in celljpj

=鈭�j鈥�E[icauses collision鈭�Placed in cellj]pj

localid="1647435990293" =鈭�j鈥�1鈭�1鈭�pji鈭�1pj

localid="1647436010058" =1鈭�鈭�j鈥�1鈭�pji鈭�1pj

03

Step 3:Final Answer

The close to last uniformity utilized that, restrictive on the itemibeing set in the cellj, itemi will cause a collision if any of the previous(i-1) items were placed in a cellj, Thus,

E[X]=m鈭�鈭�i=1m鈥�鈭�j=1n鈥�1鈭�pji鈭�1pj

Exchanging the request for the summation gives

E[X]=m鈭�n+鈭�j=1n鈥�1鈭�pjm

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Most popular questions from this chapter

Urn 1contains 5white and 6black balls, while urn 2contains 8white and 10black balls. Two balls are randomly selected from urn 1and are put into urn 2. If 3balls are then randomly selected from urn 2, compute the expected number of white balls in the trio.

Hint: LetXi = 1if the i th white ball initially in urn 1is one of the three selected, and let Xi = 0otherwise. Similarly, let Yi = 1if the i the white ball from urn 2is one of the three selected, and let Yi = 0otherwise. The number of white balls in the trio can now be written as鈭�15Xi+鈭�18Yi

Successive weekly sales, in units of 1,000, have a bivariate normal distribution with common mean 40, common standard deviation 6, and correlation .6.

(a) Find the probability that the total of the next 2 weeks鈥� sales exceeds 90.

(b) If the correlation were .2rather than .6, do you think that this would increase or decrease the answer to (a)? Explain your reasoning.

(c) Repeat (a) when the correlation is 2.

A fair die is rolled 10times. Calculate the expected sum of the10 rolls.

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(a) E[X];

(b) Var(X).

In Problem 7.9, compute the variance of the number of empty urns.
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