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October STAT/MA 28, 2011
Problem Set 26
Name:
1. Consider a pair of random variables X, Y with constant joint density on the triangle with
vertices at (0, 0), (3, 0), and (0, 3).
a. Find the expected value of the sum of X and Y , i.e., nd E(X + Y ).
b. Find the variance of X , i.e., nd Var(X ).
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2. Consider a pair of random variables X, Y with constant joint density on the quadrilateral
with vertices (0, 0), (2, 0), (2, 6), (0, 12).
a. Find the variance of X , i.e., nd Var X .
b. Find the variance of Y , i.e., nd Var Y .
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3. Let X, Y have joint density fX,Y (x, y ) = 14e2x7y for x > 0 and y > 0; and fX,Y (x, y ) = 0
otherwise.
Find the variance of the sum of X and Y , i.e., nd Var(X + Y ).
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4. X, Let Y have joint density fX,Y (x, y ) = 18e2x7y for 0 < y < x; and fX,Y (x, y ) = 0
otherwise.
Find the variance of Y .
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5. Suppose X, Y has joint density
fX,Y (x, y ) =
1
(3
9
0
x)(2 y ) if 0 x 3 and 0 y 2,
otherwise.
Find the expected value of X 2 + Y 3 , i.e., nd E(X 2 + Y 3 ).
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6. Design your own problem and solution. Create your own problem about the
expected value of a function of a continuous random variable, or about the variance of a
continuous random variable. Design your problem in such a way that you would nd it
enjoyable and also interesting (i.e., not completely trivial) if you found this problem in a
probability book. Please provide the answer for your problem too.
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