This Year Has Been Especially Wet On The Sunshine Coast Assignment

Assignment Task

Question 1:

This year has been especially wet on the Sunshine Coast where I live. Suppose the number of days, D, it rains has a Poisson distribution with λ=22.5 rainy days per month. Of course, when it rains, the actual amount of rain varies. Let R1, R2​… be the independent amounts of rain (in millimeters) on the rainy days and assume the Ri​ 's are independent of D and lognormally distributed with parameters μ= ln(6/10​) and σ= ln(10). Finally, let T=∑_iD=₁R_i be total rain next month.

a) Calculate the mean and standard deviation of T.

b) Any day with more than 100mm of rain is likely to flood my pool. What is the probability at least one such day will occur during the next month? [NOTE: The result of Question 9 from the Week 4 lab exercises may come in handy.] 

Question 2: 

When patients are discharged from the hospital, it is important to have an idea of how many are likely to be readmitted within the next month. Research on the Gold Coast has estimated the chance a patient is readmitted within a month is 20%.

a) Yesterday, the Gold Coast Hospital discharged 100 patients. Use normal approximations both with and without the continuity correction to estimate the probability 25 or more of these discharged patients are readmitted within one month.

b) If we change our focus to readmissions within a fortnight, the chances of readmission obviously decrease. A normal approximation for this situation will be less accurate than it was for part (a). Briefly explain why this is so. 

Question 3:

My daughter loves dancing. In fact, she loves it so much she takes classes 4 days a week for 42 weeks a year and therefore puts serious wear on her dancing shoes. Of course, some dance classes cause more wear than others. Suppose the wear on her shoes from each dance class has a mean of 0.09mm of shoe sole and standard deviation of 0.025mm.

a) If the soles of her dancing shoes are 15.5mm thick at the start of the year, what is the approximate chance that my daughter's shoes last out the 168 classes she takes over the year? 

b) In fact, wear is uneven between right and left shoes. Suppose wear on the right shoe from a class is normal with mean 0.1mm and standard deviation 0.03mm, while wear on the left shoe is normal with mean 0.08mm and standard deviation 0.02mm. Further, suppose the correlation between wear on each shoe on the same day is 0.8. If her right shoe sole was worn down by 0.12mm from class today, what is the probability the left shoe wear was less than 0.1mm?

Question 4: 

Consider a pdf fo(x) and suppose it has associated mgfm_o(t) = ∫^∞_∞e^tuf_o(u)du = e^t(1+t). Further, let X1, ..., Xn be a sample from a distribution with pdf

f(x;  θ) = e ^θxf_o(x)/m_o(θ) = e^θ(x−0−1)f_o(x).

[NOTE: This question can be solved without determining fo (x) explicitly.]

a) Find E(X) in terms of 8. [HINT: Find the mgf of X_i.]

b) Find the MLE of x θ

c) Find the MSE of the MLE.

d) Suppose n = 100 and Σ¹⁰⁰ _i=1X_i = 111. Find a 95% MLE-based confidence interval for θ.

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