AMS 311 Joe Mitchell
PROBABILITY THEORY: Practice Exam 3
Statistics: n= 30, µ= 64.1, median 68.5, σ= 22.9; score range: 23–99
1. (19 points) Let Xand Ybe continuous random variables with joint probability density function given by
f(x, y) = n2x+ 2yif 0 < x and x < y < 1
0 otherwise
Compute the following quantities; you may leave any integrals unevaluated, BUT you must write them completely,
with proper limits of integration, “dx”, “dy”, etc.
(a). (6 points) The marginal density, fY(y), of Y. (Be explicit about all cases.)
(b). (7 points) P(X > 0.1|Y= 0.5)
(c). (6 points) E(X|Y= 0.5)
2. (15 points) Let Ybe the lifetime, in minutes, of a lightbulb. Assume that the lightbulb has an expected lifetime
of 2 hours and that Yis exponentially distributed. At 5:00pm, the lightbulb is installed and left on. At a random
time during the lifetime of the bulb, Joe enters the room.
(a). (7 points) What time do you expect Joe will enter the room?
(b). (8 points) Find the probability that Joe enters the room after 6:40pm.
3. (16 points) An ice climber is standing on a steep slope of ice/snow, just 1 foot above a cliff. She is 3 feet below
the summit, where safety awaits her. Each time she tries to take one step upward, toward the summit, she makes
it successfully, advancing upwards by 1 foot, 30% of the time; unfortunately, 70% of the time she slides back 1 foot
behind where she started. (So, if she slides back on her first attempt to go up, she in fact falls off the cliff !) Each
step that she takes requires an amount of time, uniformly distributed between 2 minutes and 3 minutes.
For each of the following quantities, show exactly how you would compute it. Define precisely any quantities you
use!. You need not solve systems of equations, but you must be very explicit about exactly how you would obtain
the final numerical answer.
(a). (8 points) The probability that she makes it to the summit.
(b). (8 p oints) The expected number of steps she takes before she is done (she either reaches the summit and
heads home, or she falls off the cliff and never climbs again).
4. (5 points) You expect 25 p eople to show up for your party Friday night. You posted an open invitation to all of
your numerous friends to come, so now are worried that too many will show up for your small apartment to hold. If
your apartment can hold 60 people, what can you say about the probability, p, that you have to turn away people
that show up for your party?
5. (15 points) We want to find (at least approximately) the probability, p, that the average of 150 random points
independently drawn from the interval (0,1) is within 0.02 of the midpoint of the interval. Give a Central Limit
Theorem estimate for the probability p.Show your work! (Define any random variables you use!)
6. (15 points) You purchase 100 batteries for your millenium flashlight, in case the power goes out on January
1. Each battery (independently) lasts for an exponentially distributed length of time, with mean 12 hours. Your
flashlight uses only a single battery. You want to estimate the probability pthat you have enough batteries to keep
your flashlight on for more than the first 60 days of the year 2000. Obtain the best upper bound you can for p(i.e.,
find a number α, as small as possible, so that you can guarantee that p≤α).
Show your work! (Define any random variables you use!)
7. (15 points) Joe is a stamp collector, and his goal is to obtain as many of the 89 types of stamps that were produced
in the USA this year as possible. Suppose that each postage stamp is equally likely on an envelope, and Jo e obtains
a bag of 200 envelopes that people were throwing away. (Assume that each envelope has exactly one stamp on it.)
Joe sorts through the bag; each time he encounters a stamp that he does not already have, he places it into his stamp
album (which has one slot for each of the 89 different types).
What is the expected number of empty slots in Joe’s stamp album after he has sorted through the entire bag of
envelopes?
(You must show your work in order to obtain full credit. Define precisely any random variables you use.)
