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# notes - 3 Calculate V ar X 9 A rv X has pgf given by G s =...

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Test STA257 Time :2hrs Instructions : The test is out of 100 and each question is worth 10. See the end for some useful information. Please, at most 1 question/page in your booklets. No aids allowed. 1. Let X be a rv in { 0 , 1 , 2 ,... } . Show E ( X ) = n =0 P ( X > n ) . 2. Show E ( | X | ) = 0 implies P ( X = 0) = 1 . 3. Let A , B and C be independent events. Show that A and B C are also independent. 4. Let X Poisson (2) , Y Poisson (1) and Z Poisson (3) be in- dependent. Set W = X + Y + Z . Calculate P ( X = k | W = 3) for k = 0 , 1 , 2 , 3 . 5. Let X binomial (4 ,p ) , Y binomial (6 ,p ) , and Z be independent. If X + Y + Z binomial (15 ,p ) show Z binomial (5 ,p ) . 6. Let Z 1 ,Z 2 ,... be iid Bernoulli (2 / 3) and let S n = Z 1 + ··· + Z n . Let T denote the smallest n such that S n = 2 . Calculate E ( T ) . 7. Roll 2 fair die. Denote the total by T . If T < 10 you select a chip from Hat#1. Otherwise you select 3 chips without replacement from Hat#2. Hat#1 contains 3 red chips and 4 black chips while Hat#2 contains 5 reds and 2 black chips. Let A ={at least 1 read chip is selected}. Calculate P ( T < 10 | A ) . 8. Let X geometric (2
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Unformatted text preview: / 3) . Calculate V ar ( X ) . 9. A rv X has pgf given by G ( s ) = E ( s X ) = . 1 + . 4 s 4 + . 5 s 16 . Calculate E ( √ X ) . 10. Two fair 6-sided dice are rolled. Let X and Y denote the number of dots showing on each die. Let M = max { X,Y } . Calculate P ( M ≤ 2) . Information A Bernoulli ( p ) rv can only take on 1 or 0 with probabilities p and q = 1-p , respectively. The geometric ( p ) probabilities are q k-1 p,k = 1 , 2 ,... 1 + x + x 2 + ··· = 1 / (1-x ) for | x | < 1 The Poisson ( λ ) probabilities are e-λ λ k /k ! The multinomial ( N ; p 1 ,...,p k ) probabilities are N ! ( i 1 !) ... ( i k !) p i 1 1 ··· p i k k ,i 1 + ··· + i k = N . Here p 1 + ··· + p k = 1 . k = 2 yields the binomial which may also be thought of as a sum of k iid Bernoulli ( p ) rv’s. 1...
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