Homework 1 of elec 430

The current $I$ in a semiconductor diode is related to the voltage $V$ by the relation $I=e^V-1$ . If $V$ is a random variable with density function $f_V(x)=\frac{1}{2}e^{-\left|x\right|}$ for $()$∞ x ∞ , find $f(I, y)$ ; the density function of $I$ .

Suppose $X$ is a discrete random variable taking values $\{0, 1, 2, , n\}$ with the following probability mass function $p(X, k)=\begin{cases}\frac{n!}{k!(n-k)!}^k(1-)^{(n-k)} & \text{if $k=\{0, 1, 2, , n\}$}\\ 0 & \text{otherwise}\end{cases}$ with parameter $\in \left[0 , 1\right]$

Consider outcomes of a fair dice $=\{{}_1, {}_2, {}_3, {}_4, {}_5, {}_6\}$ . Define events $A=\{\colon \text{an even number appears}\}$ and $B=\{\colon \text{a number less than 5 appears}\}$ . 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 istransmitted 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 $X$ and $Y$ are each Gaussian random variables with means ${}_X$ and ${}_Y$ and variances ${}_X^2$ and ${}_Y^2$ . Assume that they are also independent. Show that $Z=X+Y$ is also Gaussian. Find the mean and variance of $Z$ .