STAT-330-1085-Midterm1_solutions

STAT-330-1085-Midterm1_solutions - 1 Let X be a...

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Unformatted text preview: 1. Let X be a Poisson-distributed random variable X with parameter 7t; i.e., its probability function is f (x) = A—je'x x. a). Find the moment-generating function for X. b) Use the mgf to show that E(X) = 7t = V(X). ' r X ‘ CY) m) g . >\ "X . 3. 6X .2 f .1 ""7 e 5:) fm 7,- a mm éé ) x. ’ t . f . ‘ '7 I KEG/l1~ ' :_ aA Aét M 66... 1 L. ,Q J7 " é ' ,n Mex/H7 w” . / _ _‘ v ,- __ . f / ; “‘ f 7 A? bi t/X) MUM we t“ O . l ‘3 fl A2 = A é ,' I/ ’ / AK? I!) f) /t 20 gm) : m Wm = I) M . l /’ W 6”) 2 26 AA” {A : flé/iz U I (2 A a +3 M . ’0 I a) Find the joint pdf for X and Y b) Find the marginal pf for X and the marginal pf for Y 0) Determine whether X and Y are independent random variables 3. Let X and Y be continuous random variables with joint density function 2 . f(x, y) = Egg-(Wm, 0 < y < x < oo a) For what values of 9 is this a pdf? b) Find f(xly) c) Find E(XIY = y) (1) Find E_(X|Y) 4. Let X and Y be continuous random variables such that X = Z and Y z 22, where the pdfonis f(z)= 1 «ff/2. J??? a) Show that C0V(X,Y) : 0. ...
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This note was uploaded on 10/16/2010 for the course STATS sta2023 taught by Professor Cullins during the Spring '10 term at Aarhus Universitet.

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STAT-330-1085-Midterm1_solutions - 1 Let X be a...

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