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Unformatted text preview: STA 4321/5325  Spring 2010
Quiz 7  April 2
Name:
There are ﬁve problems in this quiz. Each problem has exactly one correct answer.
Problem 1 The momentgenerating function of a continuous random variable X with a
probability density function f (x) is given by
(a) ∞
−∞ xf (x)dx (b) ∞
2
−∞ x f (x)dx (c) ∞x
−∞ e f (x)dx (d) ∞ tx
−∞ e f (x)dx Problem 2 Let MX (t) be the moment generating function of the random variable X . The k th
(k )
dk
derivative of MX (t) is given by MX (t) = dtk MX (t). Then the second moment of X be can
calculated by
(1) (a) MX (0)
(b) V (X )
(2) (c) MX (0)
(d) (E [X ])2
Problem 3 The momentgenerating functions are unique; that is, two random variables that
have the same momentgenerating function have the same probability distributions as well. This
statement is
(a) True
(b) False 1 Problem 4 Let X have a mixed distribution F (X ) writen uniquely as
F (X ) = cF1 (X ) + (1 − c)F2 (X )
where F1 is the distribution function of a discrete random variable X1 and F2 is the distribution
function of a continuous random variable X2 . Then E (X 2 ) is
(a) cE (X1 ) + (1 − c)E (X2 )
(b) E (X1 ) + E (X2 )
2
2
(c) cE (X1 ) + (1 − c)E (X2 )
2
2
(d) E (X1 ) + E (X2 ) Problem 5 Let X has a gamma distribution with MX (t) = (1 − βt)−α . Then E (X ) is
(a) α
(b) αβ 2
(c) α2 β
(d) αβ 2 ...
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 Spring '08
 Staff
 Probability, Probability theory, probability density function, mx, random variable X1

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