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EECE7204

Problem Set 3

Problem 1 (3.6). Let X be a Laplacian random variable with pdf

Let Y = g(X), where g(·) is the nonlinear function given as the saturable limiter

Find the distribution function FY (y).

Problem 2 (3.9). In homomorphic image processing, images are enhanced by applying nonlinear transformations to the image functions. Assume that the image function is modeled as RV X and the enhanced image Y is Y = ln X. Note that X cannot assume negative values. Compute the pdf of Y if X has an exponential density 

Problem 3 (3.10). Assume that X ∼ N (0, 1) and let Y be defined by

Compute the pdf of Y .

Problem 4 (3.18). Let X and Y be independent and identically distributed exponential RVs with

Compute the pdf of Z = X − Y .

Problem 5 (3.22). The objective is to generate numbers from the pdf shown in Figure P5 (3.22). All that is available is a random number generator that generates numbers uniformly distributed in (0, 1). Explain what procedure you would use to meet the objective.

Problem 6 (3.27). Let Z = max(X1, X2), where X1 and X2 are independent and exponentially distributed random variables with pdf

(a) Find the distribution function of Z.

(b) Find the pdf of Z.