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homework4 - Y y = 2e-2 y u y 5 Let Y = e X a Find the cdf...

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C OMER ECE 600 Homework 4 1. Let X be an exponential random variable. a. Find and plot }), {X | ( F X t x > for t a real number. b. Find and plot }). {X | ( f X t x > c. Show that }). P({X }) {X | } P({X x t x t > = > + > Explain why this is called the memoryless property. 2. Let X be the number of customers waiting for a bus. Assume that X is a geometric random variable with parameter p. Suppose that the bus can take M passengers. Find the pmf for , M) (X Y + - = where ° ± ² < = + 0 0 0 x x x x Note that Y represents the number of customers left behind. 3. Suppose that a voltage X is a zero-mean Gaussian random variable. Find the pdf of the power dissipated by an R-ohm resistor P = X 2 /R. 4. Let X be uniform on [-1,1]. Find g( x ) such that, if Y=g(X), then f
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Unformatted text preview: Y ( y ) = 2e-2 y u( y ). 5. Let Y = e X . a. Find the cdf and pdf of Y in terms of the cdf and pdf of X. b. Find the pdf of Y when X is a Gaussian random variable. 6. Find the mean and variance of the binomial random variable. 7. Show that E[X] for the random variable with cdf & ± ² < ≥-= 1 1 / 1 1 ) ( x x x x F X does not exist. 8. Let Y = Acos( & t )+c, where A is a random variable with mean m and variance ± 2 , and & and c are constants. Find the mean and variance of Y. 9. Let g(X) = ba X , where a and b are positive constants and X is a Poisson random variable. Find E[g(X)]....
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