 ## APPM 3570 / STAT 3100, Applied Probability:Exam 2

April 8, 2020

Submit your work as a PDF document on canvas before 7:00 PM. On the ftrst page, write (1) your name, (2) your section number or meeting time (10AM or 2PM) and (3)a grading table with rows for 4 problems and a total. Text books, class notes, and calculators ARE permitted. Justiftcations will be evaluated carefully so be sure tofully justify your  Correct answers without justiftcation will earn no credit.

### (1)(25 points) Random variables X, Y  have the jointdensity      应用概率代考 For this problem I strongly suggest drawing the integration regions.

(a)(5 points) FindC.

(b)(5 points) Find the marginal distribution of X,fX(x).      应用概率代考

(c)(15 points) Find the PDF of the random variable Z =Y/X.

### (2)(25 points) A bank manages 2, 000 mortgages. Mortgages default at a rate of 10 per year (you can assume this rate remains constant and that defaults can be described as a Poisson process).     应用概率代考

Every defaulted mortgage gives a loss of \$200,   Every mortgage that doesn’t default gives a yearly profit of \$2, 000. Let X be the number of mortgages that default in a year and let F be the total earnings (or losses) in a year.

(a)(5 points) Find the yearly earnings F as a function ofX.

(b)(5 points) Find the expected yearly earnings, E[F].     应用概率代考

(c)(10 points) Find the probability that earnings are negative, P (F <0).

(d)(5points) Find the probability that there are exactly five defaults in a given year, given that there have been no defaults in the first half of the year.

### (3)(25points) Suppose that Xand Yare independent Exponential random variables with mean λ= 2.

(a)(5 points) Find the joint CDF of the random variables X, Y (note that the question asks for CDF, notPDF).

(b)(10 points) Find the probability that 2X > Y.

(c)(10points) Find the expected value of X2 + XY .

### (4)(25points) Assume that the duration Tof human pregnancies has a Gaussian distribution with mean µ= 40 weeks and standard deviation σ= 2 weeks.

(a)(7 points)  A  baby  is  considered  a  “camper”  if  born  after  the  43th  week  (made  up). Find the probability p that a baby is a camper.    应用概率代考

(b)(8points) Using the Binomial distribution and your answer in (a), write down an exact expression for the probability that there will be less than 10 campers in n = 100 pregnancies. Do not evaluate the expression numerically.

(c)(10points) Using the Gaussian approximation to the Binomial (including the continuity correction), find an approximation to and evaluate it numerically using the table in the book or an online calculator.