EE465-USC-Fall08-Assignment4

EE465-USC-Fall08-Assignment4 - p and undetected with...

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EE 465 : Homework 4 Due : 9/30/2008, Tuesday in class. 1. Solve 3.5 (3.5) from textbook 2. Solve 3.22 (3.22) from textbook (Hint: Before computing E [ N ] try to compute E [ N i ], where N i is the time until the same outcome occurs i times. Also note that 1 + m + ... + m ( k 1) = m k 1 m 1 ) 3. Solve 3.27 (3.27) from textbook 4. Let X and Y be two random variables with the joint pdf f XY ( x,y ) = b x + y, 0 x 1 , 0 y 1 0 , otherwise a) Find E ( X | Y ). b) Find the pdf f Z ( z ) of Z = E ( X | Y ) Hint: The pdf of a function of a random variable Y , for example Z = g ( Y ) is given to be f Z ( z ) = s i f Y ( y i ) | g ( y i ) | where y i are real roots of the equation z = g ( y ), i.e. y i = g 1 ( z ). 5. Let Λ and X be two random variables with Λ f ( λ ) = b 5 3 λ 2 3 , 0 λ 1 0 , otherwise and f ( X | Λ = λ ) exp ( λ ). Find E ( X ). 6. Let N P ( λ ), i.e. Poisson with parameter λ be the number of photons arriving at a photodetec- tor per unit time. Each photon is detected with probability
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Unformatted text preview: p and undetected with probability (1-p ) independent of N and other photons. Let X be the number of detected photons per unit time. Thus X = ∑ N i =1 Z i , where Z i = b 1 , photon i is detected , photon i is not detected and for each N = n, { Z 1 ,Z 2 ,... ,Z n } are independent. Find the mean and variance of X . Problem 7 is an optional question, those who solve it will get extra credits. 7. Solve 3.40 (3.40) from textbook. Do part (c) for only part (a) (not for part (b)). Note : 3.2 (3.1) implies problem 3.2 from the ninth edition which is the same as 3.1 in the eighth edition....
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This note was uploaded on 08/05/2009 for the course EE 465 taught by Professor Chow during the Fall '04 term at USC.

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