EL6303
Midterm Exam Solutions
Spring 2011
1. Three types of messages arrive at a message center: high priority, denoted by the letter H, normal priority, denoted by the letter N, and low priority, denoted by the letter L. Assume that ( ) = 0.1 , ( ) = 0.4
EL6303 (Elza Erkip)
HW 1 Solution
Fall 2015
1. For any arbitrary events A, B, C with P(C ) > 0 , prove or disprove that
P( A U B)| C) 1 P( A U B U C ) .
P(C)
(Video is required.)
Solution:
Theorem: A B P( A) P(B) .
C is a subset of A U B U C P( A U B U C
EL 630: Homework 2
1. The four switches in the figure operate independently. Each switch is closed with probability p and opened with probability (1-p). (a) Find the probability that a signal at the input will be received at the output (b) Find the condit
Math 431 An Introduction to Probability Final Exam - Solutions
1.
A continuous random variable X has cdf F (x) =
a
x b
2
for x 0, for 0 < x < 1, for x 1.
(a) Determine the constants a and b. (b) Find the pdf of X. Be sure to give a formula for fX (x) th
Probability and Stochastic Processes (EL6303) December 7, 2015
NYU Polytechnic School of Engineering, Fall 2015
Instructor: Dr. Elza, Erkip
Quiz 10
Consider a WSS stochastic process X (t) with a = E (X (t) and Rxx(7') = E(X(t)X(t +
7).
1. Let Y(t) = AX (t
EL6303 HW 1 (Due 5pm, Wed, Sep 9, 2015)
1. Given:
0 P( A) 1, 0 P(B) 1.
independent.
Prove
Submit to EL6303F15@gmail.com
P( A| B) P( A| B) 1
Fall 2015
iff A, B are
(Video is required.)
2. A digital signal 1 or 0 is transmitted through a noisy channel, the
Probability and Stochastic Processes (EL6303) November 23, 2015
NYU Polytechnic School of Engineering, Fall 2015
Instructor: Dr. Elza Erkz'p
Quiz 8
Two stochastic processes X (t) and Y(t) are called jointly wide-sense stationary (W55)
if
o X(t) is WSS.
o
Probability and Stochastic Processes (EL6303) November 9, 2015
NYU Polytechnic School of Engineering, Fall 2015 7
Instructor: D7". E'lza Erkip
Quiz 6
Suppose X1,X2, . . are independent and identically distributed random variables that
are uniformly distri
EL-630: Probability Theory
Midterm Exam, March 7, 2001.
(Answer all problems, All problems carry equal weights)
1(a) Let A and B be two events such that P (A) = p1 > 0 P (B ) = p2 > 0 p1 +p2 > 1.
Show that
P (AjB ) 1 ; (1 ; p1) :
p
2
(b) A biased coin wit
7 E e EG I B E P B C Q H E Q P U 9 Q P B E V U CS H E WG P E E B Q P P H 9 E S P 9 HG E e E G I B E V U CS E Q P P 9 Q P `PGSG t 9 t C B I E Q P F U G u @ t8 q s o s p r r o @ o 6t 6 8 q @ o8 n ` t U E D G V H G m R C U C G P e U c R ` P G H U E F ` P G S
7 E e EG I B E P B C Q H E Q P U 9 Q P B E V U CS H E WG P E E B Q P P H 9 E S P 9 HG E e E G I B E V U CS E Q P P 9 Q P `PGSG t 9 t C B I E Q P F U G u @ t8 q s o s p r r o @ o 6t 6 8 q @ o8 n ` t U E D G V H G m R C U C G P e U c R ` P G H U E F ` P G S
Chapter 2
Axioms of Probability
EL6303: Introduction to Probability
Prof. Sundeep Rangan, NYU-Poly
Spring 2013
1
EL6303: Introduction to probability
Outline
Set theory, countability (not covered)
Sequences, supremum, infimum (not covered)
Axioms of pro
Chapter 4
Random Variables
EL6303: Introduction to Probability
Prof. Sundeep Rangan, NYU-Poly
Spring 2013
1
EL6303: Introduction to probability
Outline
Random variables and functions
Distributions and density functions
Gaussian distributions
Condition
Chapter 5
Functions of a Random Variable
EL6303: Introduction to Probability
Prof. Sundeep Rangan, NYU-Poly
Spring 2013
1
EL6303: Introduction to probability
Outline
Distributions of functions of random variables
Moments, means and variances
Chebyshevs
Probability and Stochastic Processes (EL6303) November 23, 2015
NYU Polytechnic School of Engineering, Fall 2015
Instructor: Dr. Elm Erkip
Quiz 9
Consider the stochastic process X (t) = Wlsin(27rft) + W2c03(27rft). Let W = (W1, W2).
We assume W1 and W2
Probability and Stochastic Processes (EL6303) November 16, 2015
NYU Polytechnic School of Engineering, Fall 2015
Instructor: Dr. Elza Erkip
Quiz 7
Consider a sequence of random variables X1, X2, . . . . We say that the sequence X con
verges to the random
EL6303 (Elza Erkip)
1.
Solution to HW 2
, where
is a step function.
Fall 2015
are positive
constants.
(1) Find constant A so that F(x) is a probability distribution function.
(2) Draw F(x).
(3) Find and draw
.
(Video is required.)
Solution:
(1)
Therefore,