5. wutao is working hard at his job in the state of Washington while finishing u

Question

5. wutao is working hard at his job in the state of Washington while finishing up the writing on his PhD dissertation. On days that he spends a full day at his office, he is not able to work on his dissertation. That happens with a probability of 0.1. On the days where he does not go to the office or is only there for part of the day, he will complete up to a maximum of 4 pages. Assume that Wutao will complete 0 to 4 full pages each day. To simplify we will round partial pages up). The probability that Wutao writes two pages is twice the probability that he writes one page. The probability he writes three pages is three times the probability that he writes one page. Finally, the probability Wutao writes 4 pages is 4 times the probability that he writes one page. Create a PMF table for the number of pages of his dissertation that Wutao completes in one day. What is the probability that Wutao will complete fewer than 2 pages in a day? On a particular day, Wutao must complete at least 2 pages. Knowing this, what is the probability he writes 4 pages? What is the expected number of pages Wutao will complete in a day? What is the standard deviation of the number of pages Wutao will complete in a day? a) b) c) d) e)

Explanation / Answer

Question 5. Pr (Will go office and no work) = 0.1

Pr( 2 page) = 2 * Pr( one page)

Pr(3 pages) = 3 * Pr( one page)

Pr( 4 page) = 4 * Pr( one page)

Let say Pr(one page) = x

so Pr(0 page) = 0.1

so Pr( 1 page) + Pr( 2 page) + Pr(3 page) + Pr( 4 page) = 0.9

x + 2x + 3x + 4x = 0.9

10x = 0.9

x = 0.09

(a) PMF table

(b) Probability that the wutao will complete fewer than 2 pages in a day = Pr (0 page) + Pr( 1 page)

= 0.10 + 0.09 = 0.19

(C) Wutao must complete at least two pages. so we have to find that

Pr ( 4 pages that day l at least 2 pages are complete) = 0.36/ (0.36 + 0.27 + 0.18) = 0.444

(D) Expected number of pages will complete in a day

E(X) = 0 * 0.1 + 1 * 0.09 + 2 * 0.18 + 3 * 0.27 + 4 * 0.36 = 2.7 pages

(e) Standard deviation of the number of pages wutao will complete = sqrt (variance)

Variance Var(X) = 0.1 * (2.7 - 0)2 + 0.09 * (2.7 - 1)2 + 0.18 * (2.7 - 2)2 + 0.27 * (2.7 - 3)2 + 0.36 * (2.7 - 4)2

= 1.71

Standard Deviation = sqrt(1.71) = 1.307

Number of Pages (X) P(X) 0 0.1 1 0.09 2 0.18 3 0.27 4 0.36

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