Assignment 8
1. Let X1 , ., Xn be random variables with the same mean and with covariance function: 2 if i = j, COV (Xi , Xj ) = 2 if|i j | = 1, 0 otherwise. Where | < 1. Find the mean and variance of
assignment 3
1. Let X be the maximum of the number of heads obtained when Carlos and Michael each ip a fair coin twice. (a) Describe the underlying space S of this random experiment and specify the pr
ASSIGNMENT 1
(1) In a specied 6-AM-to-6-AM 24-hour period, a student wakes up at time t1 and goes to sleep at some later time t2 . (a) Find the sample space and sketch in on the x y plane if the outco
Chapter 5, 6 Multiple Random Variables
ENCS6161 - Probability and Stochastic Processes
Concordia University
Vector Random Variables
A vector r.v. X is a function X : S Rn , where S is the sample space
Chapter 2 Basic Concepts of Probability Theory
ENCS6161 - Probability and Stochastic Processes
Concordia University
Specifying Random Experiments
Examples of random experiments: tossing a coin, rollin
Chapter 7 Sums of Random Variables and Long-Term Averages
ENCS6161 - Probability and Stochastic Processes
Concordia University
Sums of Random Variables
Let X1 , , Xn be r.v.s and Sn = X1 + + Xn , then
1.
2. Let A denote the event that there was a 1, and B the event that no two faces were the same.
Then,
P (B) = ( 6P3)/ 63 , P (A B) = ( 3*5P2) / 63
So,
P (A|B) = P (A B) / P (B) = (3*5P2) /
6P3
=
3.
assignment 2
1. Show that n k = n nk
2. Show that if P [A B C ] = P [A | B C ]P [B | C ]P [C ]. 3. A nonsymmetric binary communications channel is shown in the gure below. Assume the input is 0 with p
Assignment 4
1. Eight number are selected at random from the unit interval. (a) Find the probability that the rst four numbers are less than 0.25 and the last four are greater than 0.25. (b) Find the
Assignment 5
1. Let Y = A cos(t) + c where A has mean m and variance 2 and and c are constants. Find the mean and variance of Y . compare the results to those obtained in following example. Example: E
Assignment 6
1. Let X and Y have joint pdf: fX,Y (x, y ) = k (x + y ) (a) Find k . (b) Find the joint cdf of (X,Y). (c) Find the marginal pdf of X and of Y . (d) Find P [X < Y ], P [Y < X 2 ], P [X +
Assignment 7
1. Let X, Y, Z have joint pdf fX,Y,Z (x, y, z ) = k (x + y + z ) (a) Find k . (b) Find fX (x|y, z ) and fZ (z |x, y ). (c) Find fX (x), fY (y ), and fZ (z ). 2. Show that fX,Y,Z (x, y, z
Assignment 9
1. A random process is dened by Y (t) = G(t T ) where g (t) is the rectangular pulse of following gure, and T is a uniformly distributed random variable in the interval (0,1).
1
0
1
t
(a)
Assignment 10
1. Let Y (t) = X (t + d) X (t), where X (t) is a Gaussian random process. (a) Find the mean and autocovariance of Y (t). (b) Find the pdf of Y (t). (c) Find the joint pdf of Y (t) and Y
Assignment 11
1. (a) Find the autocorrelation function corresponding to the power spectral density shown in the following gure. (b) Find the total average power. (c) Plot the power in the range |f | >
Assignment 12
1. Let Mn denote the sequence of sample means form an iid random process Xn : Mn = (a) Is Mn a Markov process? (b) If the answer to part a is yes, nd the following state transition pdf: