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The december 13, 2008 final exam for the introduction to probability course (math 4510). The exam covers various topics in probability theory, including marginal distributions, expected values, poisson random variables, and conditional probabilities.
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December 13, 2008
I have neither given nor received aid on this exam.
Name:
In order to receive full credit your answer must be complete, legible and correct. Show all of your work, and give adequate explanations.
f (x, y) =
2 π
e−(x (^2) +y (^2) )/ 2 ,
and the dart board is represented by the following figure, where the disks have radius 1, 2 and 3 respectively.
3
3
(a) Compute the marginal distribution function of X.
(d) If a player gets 5 points for landing a dart in the inner gray disk, 3 points for landing a dart in the black annulus, 1 point for landing a dart in the outer gray annulus, and 0 points for missing the dart board completely, find the expected number of points a player will get after throwing a dart.
fX (t) =
1 − cos
2 πt 15
The minute after the hour that a particular person arrives at the bus stop is a uniform random variable Y on [0, 20]. Assume that X and Y are independent.
(a) Explain in words what the probability P {Y < X < Y + 5} signifies.
(b) What is the joint probability density function of X and Y (make sure to tell me the domain of this function).
(c) Compute P {Y < X < Y + 5}.
Bonus: If you were a salad dressing, what kind would you be? Explain.
