HW4_Spring11 - ELEC210 Spring 2011 Homework-4 1. The random...

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Unformatted text preview: ELEC210 Spring 2011 Homework-4 1. The random variable X has cdf as shown in the figure below: (a) What type of random variable is X? (b) Find the following probabilities: , , , , , . 2. A point is selected at random inside a square defined by . Assume the point is equally likely to fall anywhere in the square. Let the random variable Z be given by the minimum of the two coordinates of the point. (a) Find the sample space S of the coordinates of the point and the sample space of Z, (b) Show the mapping from S to . . (c) Find the region in the square corresponding to the event . (d) Find the cdf of Z. (e) Use the cdf to find: . 3. A random variable X has pdf: (a) Find c. (b) Find (c) Find . . (d) Find the mean and variance of X. 4. A limiter is shown in figure below: (a) Find an expression for the mean and variance of for an arbitrary continuous random variable X. (b) Evaluate the mean and variance if X is a Laplacian random variable with . g(x) a -a x a -a 5. A binary transmission system transmits a signal X ( -1 to send a “0” bit; +1 to send a “1” bit). The received signal is Y=X+N where noise N has a zero-mean Gaussian distribution with variance . Assume that “0” bits are three times likely as “1” bits. (a) Find the conditional pdf of Y given the input value: and . (b) The receiver decides a “0” was transmitted if the observed value of y satisfies and it decides a “1” was transmitted otherwise. What is the probability that the receiver makes an error given that a “1” was transmitted? A “0” was transmitted? Assume . (c) What is the overall probability of error? 6. (a) Plot the pdf of a Gamma random variable for and λ with Matlab (Put the three curves in one figure). (b) Plot the pdf of a Laplacian random variable for curves in one figure). with Matlab (Put the three ...
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