quiz8 - STA 4321/5325 - Spring 2010 Quiz 8 - April 9 Name:...

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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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quiz8 - STA 4321/5325 - Spring 2010 Quiz 8 - April 9 Name:...

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