Adaptive Filters

Problems and Computer Projects

PROBLEMS

Problem I.1 (Rayleigh distribution) Consider a Rayleigh-distributed random variable x with pdf given by (A.3). Show that its mean and variance are given by (A.4).

Problem I.2 (Markov’s inequality) Suppose x is a scalar nonnegative real-valued random variable with probability density function f_x (x). Show that P[x ≥ α] ≤ Ex/α.

Problem I.3 (Chebyshev’s inequality) Consider a scalar real-valued random variable x with mean and variance . Let . Apply Markov’s inequality to y to establish Chebyshev’s inequality (A.5).

Problem I.4 (Conditional expectation) Consider two real-valued random variables x and y. Establish that E [E (x|y)] = Ex. That is, show that

where S_x and S_y denote the supports of the variables x and y, respectively.

Problem I.5 (Numerical example) Assume y is a random variable that is red with probability 1/3 and blue with probability 2/3. Likewise, a? is a random variable that is Gaussian with mean 1 and variance 2 if y is red, and uniformly distributed between −1 and 1 if y is blue. Find the individual and joint pdfs of {x, y}