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 Sn = X1 + + Xn . 2. Let X1 , ., Xn be random variables
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 probabilities of its elementary events. (b) Show the mapp
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 outcome of this experiment consists of the pair(t1 , t2 ). (
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 of a random experiment. Example: randomly pick up a st
Chapter 2 Basic Concepts of Probability Theory
ENCS6161 - Probability and Stochastic Processes
Concordia University
Specifying Random Experiments
Examples of random experiments: tossing a coin, rolling a dice, the lifetime of a harddisk. Sample space: the
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 E [Sn ] = E [X1 ] + + E [Xn ] V ar[Sn ] = V ar[X1 + +
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.
4. Let X denotes the count of the numbers in the state'
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 probability p and 1 with probability 1 p. (a) Find the 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 probability that four numbers are less than 0.25 and fo
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: Expected Values of a Sinusoid with Random Phase Let Y =
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 + Y > 0.5]. 2. The random vector (X, Y ) is uniformly dis
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 ) = fZ (z |x, y )fY (y |x)fX (x). 3. Let U1 ,U2 and U3
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) Find the pmf of Y (t). (b) Find mY (t) and CY (t1 , t2
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 (t + s). (d) Show that Y (t) is a Gaussian random proce
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 | > f0 as a function of f0 > 0.
B A A
f2
f1
0
f1
f2
f
2. S
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: fMn (X |Mn1 = y ). 2. (a) Show that the following autor