1. (7 points) Answer the following questions with a very briefly justification.
a. If P{A} = P{B} = 0.3 and P{A Uß)= 0.55, are A and B independent?
b. Is the mean of 4X equal to 4 times X’s mean? Is the variance of 4X equal to 4 times X’s variance ?
c. Is it true that a continuous RV must have a continuous pdf? Can it include delta functions?
d. Give two practical interpretations of the mathematical concept of the expected value.
e. Give two reasons for the Gaussian/normal distribution to be popular in theory and application. (0 Is the sum of five arbitrary Gaussian RVs always a Gaussian RV?
(g) If F is a linear (or nonlinear) function of a RV X, can A’ and Y be uncorrelated?
1. (6 points) A truncation error X is uniformly distributed over the interval (0.10 20).
a. Write down the PDF of X.
b. What are its mean and variance?
c. Find P{0.2 x IO”20 < X < 0.6 x 10"20}. d. For a summation of 107 independent terms subject to truncation, what are the mean, variance, and PDF of its total truncation error Y1 1. (4 points) A and B shoot at the same target independently with probabilities 0.6 and 0.15, respectively, of hitting it. Find a. the probability dial the target is hit by either .4 or B b. the probability that the target is hit by A knowing that it is hit. 1. (6 points) A radar has 100 cells, each having one or no false measurement with probabilities 0.01 and 0.99, respectively, independent of the other cells. Find: a. PMF and expected number of false measurement in a cell and give the PMF’s name. b. PMF and expected number of the total false measurements of the radar over all cells and give the PMF's name. c. probability for the radar to have at least one total false measurement. 1. (4 points) Consider random variables (X, Y) having the PDF