AMS 311 Joe Mitchell
PROBABILITY THEORY: Practice Exam 2
1. (25 points) Let Xdenote the length (in meters) of one side of a square sheet of plywood. Assume that the density
function of Xis given by
f(x) = (xif 0 ≤x < 1
1/2 if 1 < x < 2;
0 otherwise.
(a). (5 points) On average, what fraction of squares of plywood have side length longer than 75cm?
(b). (6 points) Let Adenote the area (in square meters) of the sheet of plywood. Compute var (A).
(c). (14 points) Let Y= 5X. Compute the cdf, FY(y), of Y. Be very explicit! You must show the value of
FY(y) for all values of y; be careful about all cases. (You may find it helpful to sketch a plot of FY(y), but it is not
required.)
2. (15 points) Let Xdenote the lifetime of a radio, in years, manufactured by a certain company. Assume that the
lifetime is exponentially distributed, with a mean of 5 years.
(a). (5 points) What is the probability pthat a radio lasts more than 15 years?
(b). (5 points) What is the (exact) probability that, of 1000 such radios that you purchase, at least four of them
last more than 15 years? (You can leave your expression with a “p” in it, where pis the answer from part (a); you
do not need to evaluate it.)
(c). (5 points) Give an alternative expression that approximates your answer to part (b). Justify your answer:
What distribution are you using to obtain the approximation?
3. (10 points) The length of an aluminum-coated steel sheet manufactured by a certain factory is approximately
normal with mean 75 centimeters and standard deviation 2 centimeters. Find the probability that a randomly
selected sheet manufactured by this factory is between 74.2 and 75.6 centimeters.
4. (20 points) Let Xand Yhave joint density
f(x, y) = ½4/9 if 0 ≤x, and x≤y≤3−x;
0,otherwise.
(a). (7 points) Compute the marginal density of X(be explicit about all cases!).
(b). (6 points) Compute P(Y > 2X).
(c). (4 points) Compute E(XY ).
(d). (3 points) Are Xand Yindependent? (Justify your answer!)
5. (15 points) Let Xand Ybe the (indpendent) measurements of the amount of rainfall, in inches, on Long Island
on Friday the 13th, 1998. Suppose the probability density function of each measurement is given by
f(x) = n2xif 0 < x < 1
0 otherwise.
(a). Give the joint density, f(x, y) of Xand Yand sketch the support set.
(b). What is the probability that one measurement is at least twice as much as the other? (either Xis at least
twice as much as Yor Yis at least twice as much as X) (Show your work!)
6. (15 points) Suppose that Xand Yhave joint mass function as shown in the table below.
-2 0 1 3.1
-2 .1 .2 0 0
1 .1 0 .4 0
3 0 .1 0 .1
(a). (5 points) Compute P(X2> Y ).
(b). (5 points) Find the marginal mass function of Xand plot it. (be very explicit!)
(c). (5 points) Compute var(X2)
