homework6 - and variance 2 X ∼ 2 N Find the probability...

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1 EL 630: Homework 6 Note: Midterm Exam (Oct 25, 1999) 1. (a) X is a Gaussian (or Normal) r.v with mean 0 and variance . 2 σ Define . μ + = aX Y Find the characteristic function of Y ( 29 . ) ( u Y Φ (b) X ) , 0 ( 2 σ N (as before). Define . 2 X Z = Find the mean, variance and characteristic function of Z . 2. (a) When X is binomially distributed, show that ( 29 ( 29 . ) ( 1 2 1 n p q even X P - + = = (b) In a book 200 pages long, it is not unreasonable to expect 100 misprints. Find the probability that a given page will contain (i) two misprints (ii) two or less misprints (iii) two or more misprints. (Hint: Think about approximating a binomial r.v). 3. (a) (i) X is a Poisson r.v with parameter . λ Find ( 29 3 X E in terms of . λ (Hint: Use its characteristic function). (ii) Also find the characteristic function of the r.v , a X Y + = where X is as before ( 29 . ) ( λ P (b) The r.v Y is the output of a clipped rectifier as given below (also see the figure). X Y = g ( X ) = = . | | , , | | |, | ) ( a X if a a X if x X g
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Unformatted text preview: and variance . 2 X ∼ ) , ( 2 N . Find the probability density function of Y . Clipped Rectifier 2 4. (a) The discrete r.v Z has the following probability mass function ( p.m.f ) ( 29 . 1 , , 2 , 1 , , ) 1 ( < < =-= = p k p p k Z P k Find the mean and variance of Z . (b) Find the mean and variance of a Rayleigh distributed r.v X ≥ =-. , , , ) ( 2 2 2 / 2 otherwise x e x x f x X σ 5. (a) Find the characteristic function of a Gamma r.v Z ). ( )! 1 ( ) ( / 1 z U e n z z f z n n Z β---= Here n is an integer. (b) Prove or disprove the following statements. (i) ( 29 ( 29 2 2 ) ( X E X E ≥ for any random variable X . (ii) The characteristic function of a r.v can be used to compute its central moments....
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