ONLY ANSWER QUESTION 5.b 5. (4 pts each) A particular mathematics professor subm

Question

ONLY ANSWER QUESTION 5.b

5. (4 pts each) A particular mathematics professor submitted a sabbatical proposal to write a novel. He plans on taking a very mathematical approach to the writing process and so has collected much data on the genre in which he plans to write. This allows him to read many good books and justify it by saying he is doing research. His plan it to produce a finished manuscript by the end of the sabbatical period (a) Within the genre, the professor discovers the average length of the novel is approximately normally distributed with a mean of 500 pages with a standard deviation of 120 pages. The professor randomly picks up another book for more "research." What are the chances that this new book is between 450 and 475 pages long? (b) The math professor plans on sprinkling mathematical references through the work. Sup- pose in a six chapter section, the professor has six mathematical references he wants to use. These can be placed anywhere, with each reference being equally likely to be placed in any one of the six chapters. What is the probability that some chapter has two or more references? (c) The number of type-Os on a page of the professor's manuscript is a poisson random te the chances of a random page of the manuscript variable with mean of 6. Calcula having 8 or more type-Os. (d) Suppose the time it takes the professor to finish the novel is exponentially distributed with a mean of 1200 hours. The professor has already spent 1000 hours on the manuscript Ho w much additional time ca n he expect to spend writing (e) The time, T, the professor spends writing on a particular day is uniformly distributed between 4 and 8 hours. The number of pages written in a day, P, depends on the time spent writing, t, and is a poisson random variable with a variance of 5t. Find the expected number of pages written in a day

Explanation / Answer

Here first we will calculate the number of possibilities that a chapter. Lets say each chapter contain xi references x1,x,2,x3,x4,x,5,x6

where x1 + x2 + x3 + x4 + x5 + x6 = 6

and xi E [0, 6]

Here n = 6 and k = 6

Total number of solution here for this combination = (n+k-1)C(n-1) = (6 + 6 -1)C(6-1)   = 11C5 = 462

Here we have to find that some chapter has two or more refences that means it can be calculated by removing the possibilites that there is one reference from each chapter. in this case only each chapter has only one reference. Otherwise in any other case, there will be chance that it will have equal or more than 2 references.

Number of possbilities when one reference from each chapter = 1

Pr(2 or more references in one chapter) = 1 - Pr(Each chapter has one reference each) = 1 - 1/462 = 461/462

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