Probability Theory and Statistics

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 .

1. (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)?
2. (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:

6e−(2x+3y) x≥0,y≥0
fX,Y(x,y)= 0 otherwise

1. (a) Are X and Y independent or not?
2. (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 T0) 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.

1. (a) Determine the conditional PDF fX |H (x).
2. (b) Calculate the conditional probabilities P(T+|H) and P(T−|H).
3. (c) Find P(H|T−), the probability that a person is actually healthy, given the event of a negative test
   result.
4. (d) If the test is ambiguous (T0), 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(T0) for a randomly selected person.