20061ee131A_1_131a4

20061ee131A_1_131a4 - k = 0 , 1 , 2 , 3 with n = 10 and p =...

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EE 131A Homework #4 Winter 2006 Due February 16th K. Yao Read Leon-Garcia, pp. 99–118 1. The Rayleigh random variable has c.d.f. F R ( r ) = ( 0 , r < 0 1 - e - r 2 / 2 σ 2 , r 0 , Find P [ σ R 2 σ ] and P [ R > 3 σ ]. Find the p.d.f. of this r.v. 2. A r.v. X has pdf of f X ( x ) = ( cxe - x , x 0 0 , x < 0 , a. Find the constant c. b. Find the cdf of X. c. Find the probability that X > 0. 3. Suppose X is the number of hours a component part will last and X has a pdf given by f ( x ) = e - x , x 0 . a. Find the probability that a component lasts at least 6 hours. b. Suppose a machine contains 3 components. Find the probability that they all last at least 6 hours, assuming that the components function independently of one another. Hint: Let X be the number of hours the first component lasts; Y be the number of hours the second component lasts; Z the number of hours the third component lasts. 4. Suppose X has a pdf f ( x ) and a cdf F ( x ). What can one conclude about f ( x ) and F ( x ) if a. X is never between 2 and 6. b. X is always between 2 and 6. c. X is never 3. d. X is never 4. 5. If X is a Poisson r.v. with P ( X = 0) = e - 2 , find P ( X > 2). 6. Compare the Poisson approximation to the binomial probabilities for
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Unformatted text preview: k = 0 , 1 , 2 , 3 with n = 10 and p = 0 . 1; n = 20 and p = 0 . 05; n = 100 and p = 0 . 01. 7. Let X denote the number of hours a student studies during a randomly selected day. Suppose the probability law is specied by the cdf given by F ( x ) = , x &lt; 1 8 x + 1 8 , x &lt; 1 1 2 , 1 x &lt; 2 1 8 x + 1 2 , 2 x &lt; 4 1 , x 4 . a. Sketch the cdf. b. Find the probability that the student: (i) studies exactly 2 hours; P ( X = 2) (ii) studies exactly 3 hours; P ( X = 3) (iii) studies; that is, P ( X &gt; 0) (iv) studies more than 2 hours; P ( X &gt; 2) (v) studies less than 2 hours; P ( X &lt; 2) (vi) studies between 1 and 3 hours; P (1 &lt; X &lt; 3) (vii) studies more than 2 hours given that he does study; P (2 &lt; X | X &gt; 0) (viii) studies less than 3 hours given that he studies more than 1 hour; P ( X &lt; 3 | X &gt; 1). 2...
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20061ee131A_1_131a4 - k = 0 , 1 , 2 , 3 with n = 10 and p =...

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