EL6303 (Elza Erkip)
HW 3
Fall 2015
1. Let
(a) Find B so that
represents a probability mass function.
(b) Find
(c) Find
(d) Find
.
(e) Let
. Find the conditional probability mass function
.
(Video is r
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
EL6303
Solutions to HW 4
Summer 2015
1. Show that if X 0 and Ecfw_X , then Pcfw_X .
Clue: Understand the proof of Tchebycheff Inequality
(Video is required.)
Solution: If
for
f ( x) 0
Pcfw_ X
Proo
EL6303
Solutions to HW 1 Summer 2015
1. Prove or disprove that ( a) A B A B A ;
(b) ( A B ) AB AB B A .
Solution : ( a ) According to De Morgan's Law A B AB, we have
A B AB AB ,and A B AB
then , A B A
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 i
EL6303 (Elza Erkip)
HW 1
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.)
2.
Given: 0 < P( A) <1, 0 < P(
EL6303
1. Find
Solution to HW 2
Summer 2015
. (Do not just use integral formula, prove the formula.)
e x dx
Solution:
,
x
x
y
( x y )
2
I e dx I e dx e dy e
dxdy
,
,
x r cos y r sin dxdy rdrd
2
2
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 a
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, den
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) Fin
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 j
EL6303 Probability and Stochastic Processes
XK Chen, NYU/ECE
Introduction
Probability and Stochastic processes is an interesting branch of
mathematics that deals with measuring or determining quantita
EL6303
Solution to HW 2
Fall 2017
1. F ( x) A(1 ebx )u( x a) , where u( x) is a step function. a and b are positive
constants.
(1) Find constant A so that F(x) is a probability distribution function.
EL6303
HW 6 Solutions
Fall 2017
1. X and Y have joint density function
f ( x, y)
XY
1 xy
A
, | x |1,| y |1; zero,otherwise.
(1) Find A so that the above defined is a valid joint density function.
(2)
EL6303
Solution to HW 3
Fall 2017
1. Let P( X k ) Bkp k 1, k 1,2,., ; 0 p 1.
(a) Find B so that P( X k ) represents a probability mass function.
(c) Find Ecfw_X 2.
(b) Find Ecfw_X .
(d) Find Ecfw_X 2
EL6303
HW 4
Solution
Fall 2017
1. Find and draw FY ( y ) and fY ( y ) in terms of FX ( x) and f X ( x) .
2, for x 0
0, for 0 x 1
y g ( x)
x 1, for 1 x 2
1, for x 2
f X ( x)
x
(Video is required.)
EL6303
Solution to HW 5
Fall 2017
1. X and Y are independent with f X ( x) u( x 1) u( x 2), fY ( y) u( y) u( y 1) .
Z X Y . Find f Z ( z) by
f X ( ) fY ( z )d
(1) Convolution Method. f Z ( z) f X ( z)
PROBLEMS
CHAPTER 11
1. Consider an economy in which the marginal propensity to consume is .9, prices are
constant, the multiplier is 10, G is initially 1000, taxes are autonomous (not related to
incom
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, . . .
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)
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 ran
Lecture 2
Chapter 4 The Concept of a Random Variable
4.1 Introduction
Example
In the fair-die experiment, if we assign to the six outcomes fi the
numbers X( fi ) 10i, i.e. X( f1) 10, . X( f 6 ) 60,
th
EL6303
HW 1 Solution
Fall 2017
1. For any arbitrary events A, B, C with P(C ) 0 , prove or disprove that
P( A B)| C ) 1 P( A B C ) .
P(C)
Solution:
Theorem: A B P( A) P(B) .
C is a subset of A B C P(
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 (
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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
2.5 Social experiment, Part I. A social experiment conducted by a TV program questioned
what people do when they see a very obviously bruised woman getting picked on by her
boyfriend.On two different
EL6303
HW 1
Fall 2017
1. For any arbitrary events A, B, C with P(C ) 0 , prove or disprove that
P( A B)| C ) 1 P( A B C ) .
P(C)
2.
B
A
A B
1
Given: 0 P( A) 1, 0 P(B) 1. Prove P( A| B) P( A| B) 1 iff
EL6303
HW 2
Fall 2017
1. F ( x) A(1 ebx )u( x a) , where u( x) is a step function. a and b are positive
constants.
(1) Find constant A so that F(x) is a probability distribution function.
(2) Draw F(x