homework4

# homework4 - 4 Suppose that a voltage X is a zero-mean...

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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. A random variable X is said to be a geometric random variable if 1 P(X ) 1,2,3, k k pq k - = = = where p , q >0 and p + q =1. a. Show that for any natural numbers m and n , P(X | X ) P(X ) m n m n > + > = > This is known as the memoryless property of a geometric random variable. b. Show that the converse of part a is also true, i.e., if X is a positive integer- valued random variable satisfying the memoryless property for any two natural numbers m and n , then X is in fact a geometric random variable. 3. 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.

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Unformatted text preview: 4. 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. 5. Let X be uniform on [-1,1]. Find g( x ) such that, if Y=g(X), then f Y ( y ) = 2e-2 y u( y ). 6. 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. 7. Find the mean and variance of the binomial random variable. 8. Show that E[X] for the random variable with cdf 1 1/ 1 ( ) 1 X x x F x x-≥ & = ± < ² does not exist. 9. 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. 10. 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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homework4 - 4 Suppose that a voltage X is a zero-mean...

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