ENGR 3341 Probability Theory and Statistics Homework #9, due 3/26 Prof. Gelb Warm-up problems from textbook (answers on eLearning): Section 5.4 problems 20, 21, 25, 28, 32 Problems: 1. Suppose your course grade depends on two test scores: X1 and X2 . Each score is a Gaussian(µ = 74, σ = 16) random variable, independent of the other. (a) With equal weighting, grades are determined by Y = X1 /2 + X2 /2. An “A” grade requires Y ≥ 90; what is P(A) = P(Y ≥ 90)? Useful fact: the sum of two independent gaussian random variables is also a gaussian random variable, so you only need to determine µY and σY . (b) A student proposes that only the better of the two exam scores M = max(X1 , X2 ) should be used to determine the course grade. The professor agrees; what is P(A) = P(M ≥ 90)? (c) In a class of 100 students, what is the expected increase in the number of A’s awarded due to this change in policy? 2. Y is the exponential(0.2) random variable. Given A = {Y < 2}, calculate fY |A (y) and E[Y |A]. 3. Random variables X and Y have joint PDF:  −(2x+3y) 6e fX,Y (x, y) = 0 x ≥ 0, y ≥ 0 otherwise (a) Are X and Y independent or not? (b) Let A be the event X +Y ≤ 1. Calculate the conditional joint PDF fX,Y |A (x, y). Are X and Y independent, given A? 4. A test for diabetes is a measurement of a person’s blood sugar level, X, following an overnight fast. For a healthy person, a result in the range 70—110 mg/dl is considered normal. A “positive” for diabetes, (event T + ) is X ≥ 140; a “negative” (event T − ) is X ≤ 110; the test is considered ambiguous (event T 0 ) if 110 < X < 140. For a healthy person (event H) X is the Gaussian random variable with µ = 90 and σ = 20. For someone with diabetes (event D), X is Gaussian with µ = 160 and σ = 40. The probability that a randomly chosen person has diabetes is P(D) = 0.10, and P(H) = 0.90. (a) Determine the conditional PDF fX|H (x). (b) Calculate the conditional probabilities P(T + |H) and P(T − |H). (c) Find P(H|T − ), the probability that a person is actually healthy, given the event of a negative test result. (d) If the test is ambiguous (T 0 ), it is repeated until either a positive or negative result is obtained. Calculate the expected number of tests required to obtain an unambiguous result for a randomly selected person. Hint: first calculate the probability P(T 0 ) for a randomly selected person. 1