. Librarians in busy libraries can reshelf quite a few books in a day. Assume th
ID: 3131505 • Letter: #
Question
. Librarians in busy libraries can reshelf quite a few books in a day. Assume that for each librarian 4 books per day get placed on the wrong shelf on average. Assume that correct shelving of any books is independent of the others. a) Let B be the number of books that a librarian places on the wrong shelf on a particular day. What are the distribution, parameter(s) and support of B? b) What is the probability that 10 books get placed on the wrong shelf on a particular day? c) What is the probability that a librarian placed 4 books on the wrong shelf on Monday and 6 books on the wrong shelf on Tuesday? d) Let M be the number of times that a book gets placed on the wrong shelf in the month of March. What are the distribution, parameter(s) and support of M? The library was open every day in March. e) The library has a policy that if a librarian puts 100 books on the wrong shelf in a month they will be reprimanded. What is the probability that a librarian has incorrectly shelved between 99 books and 101 books (inclusive) in March? f) In the first 15 days of March, the librarian placed 48 books on the wrong shelf. What is the probability the librarian will have incorrectly shelved 100 books by the end of March?
Explanation / Answer
a) Let B be the number of books that a librarian places on the wrong shelf on a particular day. What are the distribution, parameter(s) and support of B?
It is a Poisson distribution of mean u = 4. The x values can take any none-negative integer value, 0,1,2,3... .
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b) What is the probability that 10 books get placed on the wrong shelf on a particular day?
Note that the probability of x successes is
P(x) = u^x e^(-u) / x!
where
u = the mean number of successes = 4
x = the number of successes = 10
Thus, the probability is
P ( 10 ) = 0.005292477 [ANSWER]
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c) What is the probability that a librarian placed 4 books on the wrong shelf on Monday and 6 books on the wrong shelf on Tuesday?
On Monday:
Note that the probability of x successes is
P(x) = u^x e^(-u) / x!
where
u = the mean number of successes = 4
x = the number of successes = 4
Thus, the probability is
P ( 4 ) = 0.195366815
On Tuesday:
Note that the probability of x successes is
P(x) = u^x e^(-u) / x!
where
u = the mean number of successes = 4
x = the number of successes = 6
Thus, the probability is
P ( 6 ) = 0.104195635
Hence,
P(4 on monday 6 on tuesday) = 0.195366815*0.104195635 = 0.020356369 [ANSWER]
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d) Let M be the number of times that a book gets placed on the wrong shelf in the month of March. What are the distribution, parameter(s) and support of M? The library was open every day in March.
By central limit theorem, the distribution is approimately normal, with mean (As there are 31 days for march)
u(sum) = u*31 =4*31 = 124
and standard deviation (as sigma = sqrt(u) for Poisson)
sigma(sum) = sigma*sqrt(n) = sqrt(4)*sqrt(31) = 11.13552873
Also, the support is also any nonnegative integer, {0,1,2,3...}.
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