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Unformatted text preview: STA 4321/5325  Spring 2010
Quiz 8  April 9
Name:
There are ﬁve problems in this quiz. Each problem has exactly one correct answer.
Problem 1 Let f (x, y ) denote the joint probability density function of two continuous random
variables X and Y , and fX (x) denote the marginal probability density function of X . Then it is
always true that
(a) fX (x) = ∞
−∞ f (x, y )dx (b) fX (x) = ∞
0 f (x, y )dy (c) fX (x) = f (x, 0)
(d) fX (x) = ∞
−∞ f (x, y )dy Problem 2 Let X and Y be two discrete random variables with the following joint probability
mass function: X p(x, y )
0
1
2 0
1/9
2/9
1/9 Y
1
2/9
2/9
0 2
1/9
0
0 Then,
(a) P (X = 1) = 4
81 (b) P (X = 1) = 2
9 (c) P (X = 1) = 4
9 (d) P (X = 1) = 1
Problem 3 Let X and Y be two continuous random variables with joint probability density
function f (x, y ), and marginal density functions fX (x) and fY (y ) respectively. Then X and Y are
said to be independent if f (x, y ) = fX (x)fY (y ) for every x ∈ R, y ∈ R. This statement is
(a) True
(b) False 1 Problem 4 Let X and Y be continuous random variables taking positive values, with joint
probability density function given by f (x, y ) = e−(x+y) for every x > 0, y > 0. It can be derived
that the marginal probability density function of Y is given by fY (y ) = e−y for y > 0. Then, the
conditional probability density function of X given Y = 7 is given by
(a) fX Y =7 (x) = e−7 for every x > 0
(b) fX Y =7 (x) = e−(x+7) for every x > 0
(c) fX Y =7 (x) = e−x for every x > 0
(d) fX Y =7 (x) = e−(y+7) for every x > 0
Problem 5 Let X and Y be two independent Bernoulli random variables. Let P (X = 0) =
1
and P (Y = 0) = 3 . Then,
(a) P (X = 0, Y = 1) = 4
9 (b) P (X = 0, Y = 1) = 2
9 (c) P (X = 0, Y = 1) = 3
9 (d) P (X = 0, Y = 1) = 1
9 2 1
3 ...
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 Spring '08
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 Probability, Probability theory, probability density function

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