Random VectorsExtend from pairs to length 𝑛• pmf, pdf and cdf• Independence• Expectation – Covariance matrix – Cross-correlation

• Functions – n-to-1– n-to-n– Linear

• Conditioning• Gaussian case

8.1.4 X = [X1 X2 X3]′ has PDF

LetU = [X1 X2]′, V = [X1 X3]′ andW = [X2 X3]′. Find the marginal PDFs fU(u), fV(v) and fW(w).

8.2.3 The random vectorX has PDF

where a = [1 2 3]′. Are the components ofX independent random variables?

8.2.4 The PDF of the 3-dimensional random vectorX is

Are the components ofX independent random variables?Find the marginal PDFs fx1(x1), fx2(x2), and fx3(x3).

8.3.3 In an automatic geolocation system, a dispatcher sends a message to six trucks in a fleet asking their locations. The waiting times for responses from the six trucks are iid exponential random variables, each with expected value 2 seconds.(a) What is the probability that all six responses will arrive within 5 seconds?(b) If the system has to locate all six vehicles within 3 seconds, it has toreduce the expected response time of each vehicle. What is the maximum expected response time that will produce a location time for all six vehicles of 3 seconds or less with probability of at least 0.9?

8.3.4 Let X1, . . . , Xn denote n iid random variables with PDF fX(x) and CDF FX(x). What is the probability P[Xn = max{X1, . . . , Xn}]?

8.4.9 The 4-dimensional random vector Y has PDF

Find the expected value vector E[Y] and the covariance matrix CY.

8.5.1 X is the 3-dimensional Gaussian random vector with expected value µX = [4 8 6]′ and covariance

Calculate(b) the PDF of the first two components ofX, fX1,X2(x1, x2),

(c) the probability that X1 > 8.

8.5.3 Given the Gaussian random vectorX in Problem 8.5.1, Y = AX + b, where

and b = [−4 −4]′. Calculate(a) the expected value µY,(b) the covariance CY,(d) the probability that −1 ≤ Y2 ≤ 1.

5.10.8 In a race of 10 sailboats, the finishing times of all boats are Gaussian random variables with expected value 35 minutes and standard deviation 5 minutes. (a) What is the probability that the winning boat will finish the race in less than 25 minutes?(b) What is the probability that the last boat will cross the finish line in more than 50 minutes?(c) Given this model, what is the probability that a boat will finish before it starts (negative finishing time)?8.6.2 To account for common wind and tides, drop the iid assumption and add correlation coefficient ρ = 0.8.(a) What is the covariance matrix of X = [X1 ... X10]′?

(b) Let

What are the expected value and variance of Y? What is P[Y ≤ 25]?8.6.4 LetW denote the finish time of the winning boat. Can we find [W ≤ 25], the probability that the winning boat finishes in under 25 minutes? Simulate?