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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.
(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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- Spring '11