AMCS 241, Probability and Random Processes, 1st Semester 2016-2017
Assignment #11, due on Sunday, November 27th
1. a. A finite DTMC with positive-recurrent aperiodic
states has a TPM that is doubly st
AMCS 310: Assignment 3
Due on Oct 9, 2017
1. Download the dataset air.txt. This dataset contains measurements of the ambient concentration (in g/m3 ) of the pollutant sulphur dioxide (SO2 ) for large
AMCS 310 Homework 2
Submitted by Mary Lai Salvaa
ID No. 156071
September 24, 2017
1. Show that (1 f )pq/(n 1) is an unbiased estimator of Var(p), the variance of the sample
proportion from a sample of
AMCS 310 Homework 3
Submitted by Mary Lai Salvaa
ID No. 156071
October 9, 2017
1. Air Dataset
a. Perform and briefly summarize an exploratory data analysis of these data.
Based on the histogram plots,
STAT 220: Probability and Statistics, Fall 2017
Homework 5
[A] Problems for practice:
Exercise A1: Textbook 5.8.14
Exercise A2: Textbook 5.8.15
[B] Problems due October 4, 2017, in class:
Exercise B1:
STAT 220: Probability and Statistics, Fall 2017
Homework 3
[A] Problems for practice:
Exercise A1: Textbook 3.8.17
Exercise A2: Textbook 3.8.24
[B] Problems due September 20, 2017, in class:
Exercise
STAT 220: Probability and Statistics, Fall 2017
Homework 2
[A] Problems for practice:
Exercise A1: Textbook 2.14.11
Exercise A2: Textbook 2.14.17
[B] Problems due September 13, 2017, in class:
Exercis
AMCS 243 Homework 4
Submitted by Mary Lai Salvaa
ID No. 156071
September 26, 2017
1. Let X Exp(). Find P (|X X | kX ) for k > 1. Compare this to the bound you get
from Chebyshevs inequality.
Let X Exp
AMCS 243 Homework 3
Submitted by Mary Lai Salvaa
ID No. 156071
September 19, 2017
1. Let X be continuous random variable
with CDF F . Suppose that P (X > 0) = 1 and that
R
E[X] exists. Show that E[X]
AMCS 310: Assignment 2
Due on Sep 25, 2017
1. Show that (1 f )pq/(n 1) is an unbiased estimator of the V ar(p), the variance of the
sample proportion from a sample of size n from a finite population o
BiSSon Pr(m ASSuv-ftons
P. Poisson frown 0% hill K70 is a Counfthj Process Na'), t7/o, ank
Values in The, Ed: [01 JZIWH such MT:
0. INC) :0 Odd N(cfw_:) 'Is nonJcCreasiv, cfw_-9 if Gin, Mgr/Mt)-
b- It
AMCS 241, Probability and Random Processes, 1st Semester 2016-2017
Assignment #2, due on Sunday, September 4th
1. a. Let , , , , , . be independent RVs. The RV takes positive integer values and has
me
AMCS 241, Probability and Random Processes, 1st Semester 2016-2017
Assignment #10, due on Sunday, November 20th
1.
Obtain the joint CDF and PDF of the residual and current lifetimes, i.e., () and , of
AMCS 241, Probability and Random Processes, 1st Semester 2016-2017
Assignment #9, due on Sunday, November 13th
1. Obtain the following distributions/probabilities for a Poisson process with rate :
a.
AMCS 241, Probability and Random Processes, 1st Semester 2016-2017
Assignment #5, due on Sunday, October 2nd
1. a. In HW2, Problem 8, if = , what is the PGF of ? Find the PMF of using Matlab instructi
AMCS 241, Probability and Random Processes, 1st Semester 2016-2017
Assignment #8, due on Sunday, October 23rd
1. a. Prove that if
and
> =
Note that for RV and > 0,
, then
= . Is the function
.
AMCS 241, Probability and Random Processes, 1st Semester 2016-2017
Assignment #1, due on Sunday, August 28th
1. a. Consider events and . Find if = . and = . .
b. Consider , , . Given that = . , = . an
AMCS 241, Probability and Random Processes, 1st Semester 2016-2017
Assignment #7, due on Sunday, October 16th
1. a. Prove that log when > and = .
b. If = and = , prove that > .
c. Suppose that you are
AMCS 241, Probability and Random Processes, 1st Semester 2016-2017
Assignment #6, due on Sunday, October 9th
=
1. Consider independent RV's
where is a Laplacian RV with PDF
=
exp ,
=
We are intere
AMCS 241, Probability and Random Processes, 1st Semester 2016-2017
Assignment #3, due on Sunday, September 18th
1. Let be the PDF of the sum of i.i.d. positive RV's. Let be their CDF. That is, if thes
STAT220 Probability and Statistics
Homework #2 Solutions
Exercise A1 - 2.14.11 - 0 pts:
Suppose we toss a coin once and let p be the probability of heads. Let X denote
the number of heads and let Y de
STAT220 Probability and Statistics
Homework #1 Solutions
Exercise A1 - 1.10.10 - 0 pts:
You have probably heard it before. Now you can solve it rigorously. It is called the
Monty Hall Problem. A prize
AMCS 243 Homework 1
Submitted by Mary Lai Salvaa
ID No. 156071
August 29, 2017
1. There are three cards. The first is green on both sides, the second is red on both sides, and
the third is green on on
1
Probability
1.1
Introduction
Probability is a mathematical language for quantifying uncertainty. In this
Chapter we introduce the basic concepts underlying probability theory. We
begin with the samp
6
Models, Statistical Inference and
Learning
6.1
Introduction
Statistical inference, or "learning" as it is called in computer science, is the
process of using data to infer the distribution that gene
4
Inequalities
4.1
Probability Inequalities
Inequalities are useful for bounding quantities that might otherwise be hard
to compute. They will also be used in the theory of convergence which is
discus
9
Parametric Inference
We now turn our attention to parametric models, that is, models of the form
.;y =
cfw_f(x; 8):
8
E
8
(9.1)
where the 8 c ~k is the parameter space and 8 = (8 1 , . , 8k ) is the
An Introduction to
Robust Estimation with
R Functions
Ruggero Bellio
Department of Statistics, University of Udine
[email protected]
Laura Ventura
Department of Statistics, University of Pad
Op and Op
In statisics, probability and machine learning, we make use of 0p and Op notation.
Recall rst, that on = 0(1) means that an > 0 as n + 00. on = 0(1) means that
(tn/b = 0(1).
on = 0(1) means
3
Expectation
3.1
Expectation of a Randorn Variable
The mean, or expectation, of a random variable X is the average value of
x.
3.1 Definition. The expected value, or mean, or first moment, of
X is de
14
Multivariate Models
In this chapter we revisit the Multinomial model and the multivariate Normal.
Let us first review some notation from linear algebra. In what follows, x and
yare vectors and A is