HW7_Spring11_Sol

HW7_Spring11_Sol - ELEC210 Spring 2 011...

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Unformatted text preview: ELEC210 Spring 2 011 Homework-7-Solutions 1. Let X , X be the jointly Gauss sian random variables with mean vecto r and covariance matrix given by: 9/5 2/5 1 , 1 2/5 6/5 (a) Find the pdf of in matrix notati on. (b) Find the pdf of using the quadr atic expression in the exponent. (c) Find the marginal pdfs of X and X . d Solutions (20’) (a) According to f | the matrix notation is f ex p 1/5 , det 9/1 0 3/5 1/5 1 , 1 and | √ ex p 1 (b) 3 5 Therefore, f (c) According to f Therefore f 2. Let X , … , X 1 √ √ 2 5 1 x x 1 1 m 1 9 10 1 1 exp exp 1 1 1 exp 1 where , √ ,f 1 , exp x 1 be the random variable s with the same mean μ and with cova riance function: COV X , X where | | 2 if i j σ if |i j| 1 ρσ 0 otherwise 1. Find the mean and var iance of Solutions (10’) ⋯ . 3. Suppose that 30% of voters are in fav or of certain legislation. A large numb er n of voters are polled and a relative frequency estima te f for the above proportion is obtained. Use the following formula P |M μ| ε 1 to determine how many voters should be polled in order that the probabilit is at least 0.90 ity that f differs from 0.30 by less th an 0.02. Solutions (10’) P |M μ| where ε 1 and thus P |M μ μ μ| 0.21, μ ε 0.3, 0.1 0.02 n 4. A discrete-time random process is de fined by X s , for n random from the interval 0,1 . (a) Sketch some sample paths of the p rocess. (b) Find the cdf of X . (c) Find the joint cdf for X and X . (d) Find the mean and auto-covarianc e functions of X . Solutions (20’) 5 250 0, where s is selected at 5. Let g t be the rectangular pulse sho wn in the figure. The random proces s X t is defined s as Xt Ag t Where A assumes the values 1 wit h equal probability. (a) (b) (c) (d) Find the pmf of X t . Find m t . Find the joint pmf of X t and X t Find C t, t d , d 0. Solutions (20’) d. For t ∉ 0,1 , t d ∈ 0,1 , P x t 0, x t d 1 6. A student uses pens whose lifetime is an exponent random variable with m ean 2 weeks. Use s the central limit theorem to determin e the minimum number of pens he s hould buy at the beginning of a 15-week semester, so that with probability 0.99 he does no t run out of pens during the semester. Hint: 2.3263 0.99 Solutions (20’) Denote as the lifetime of the ith pe n, and ⋯ pens. ∙2 2 By independence between the lifetime of different pens, VAR Assuming VAR 4 ∙2 is approximately Gauss ian, and by applying CLT: 15 From hint: 17.12 √ √ √ √ 0.99 2.3263 3.28 and this student sh ould buy at least 18 pens. as the total lifetime of the ...
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This note was uploaded on 10/16/2011 for the course ELEC 308,315,10 taught by Professor Prof.shenghuisong during the Spring '11 term at CUHK.

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