The current II in a semiconductor diode is related to the voltage VV by the relation I=ⅇV−1 I V 1 . If VV is a random variable with density function f V x=12ⅇ-|x| f V x 1 2 x for -∞<x<∞ x , find fIy f I y ; the density function of II.

Suppose XX is a discrete random variable taking values 012…n 0 1 2 … n with the following probability mass function pXk=n!k!n−k!θk1−θn−kifk=012…n0otherwise p X k n k n k θ k 1 θ n k k 0 1 2 … n 0 with parameter θ∈01 θ 0 1

Consider outcomes of a fair dice Ω= ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 Ω ω 1 ω 2 ω 3 ω 4 ω 5 ω 6 . Define events A={ω|an even number appears} A ω an even number appears ω and B={ω|a number less than 5 appears} B ω a number less than 5 appears ω . Are these events disjoint? Are they independent? (Show your work!)

This is problem 3.5 in Proakis and Salehi.

An information source produces 0 and 1 with probabilities 0.3 and 0.7, respectively. The output of the source is transmitted via a channel that has a probability of error (turning a 1 into a 0 or a 0 into a 1) equal to 0.2.

Suppose XX and YY are each Gaussian random variables with means μ X μ X and μ Y μ Y and variances σ X 2 σ X 2 and σ Y 2 σ Y 2 . Assume that they are also independent. Show that Z=X+Y Z X Y is also Gaussian. Find the mean and variance of ZZ.