Stat W4109: Midterm II
Fall 2013
Time Limit: 2.5 hours
Name (Print):
Student UNI:
Signature:
This exam contains ? problems. Answer all of them. Point values are in parentheses. You must
show your work to get credit for your solutions - correct answers wit
Stat W4109
Fall 2013
Time Limit: 2.5 hours
Name (Print):
Student UNI:
Signature:
This exam contains 7 problems. Answer all of them. Point values are in parentheses. You must
show your work to get credit for your solutions - correct answers without work wi
Stat W4109
Fall 2013
Time Limit: 2.5 hours
Name (Print):
Student UNI:
Signature:
This exam contains 7 problems. Answer all of them. Point values are in parentheses. You must
show your work to get credit for your solutions - correct answers without work wi
Stat W4109: Practice Final
Fall 2013
Time Limit: 2 hours 45 minutes
Name (Print):
Student UNI:
Signature:
This exam contains 8 problems. Answer all of them. Point values are in parentheses. You must
show your work to get credit for your solutions - correc
Stat W4109: Final
Fall 2013
Time Limit: 2 hours 45 minutes
Name (Print):
Student UNI:
Signature:
This exam contains 7 problems. Answer all of them. Point values are in parentheses. You must
show your work to get credit for your solutions correct answers w
Stat W4109: Practice Final
Fall 2013
Time Limit: 2 hours 45 minutes
Name (Print):
Student UNI:
Signature:
This exam contains 8 problems. Answer all of them. Point values are in parentheses. You must
show your work to get credit for your solutions - correc
Stat W4109
Fall 2013
Time Limit: 2.5 hours
Name (Print):
Student UNI:
Signature:
This exam contains 7 problems. Answer all of them. Point values are in parentheses. You must
show your work to get credit for your solutions - correct answers without work wi
Stat W4109: Practice problems for Midterm-I
1. The probability that each specic child in a certain family will inherit a certain disease
is p. If it is known that at least one child in a family of n children has inherited the
disease, what is the expected
Introduction to Probability Lectures 9 & 10
October 5, 2013
In this lecture we introduce a number of approximation results.
1
Probability inequalities
There is an adage in probability that says that behind every limit theorem lies a
probability inequality
W4109: PROBABILITY & STATISTICAL INFERENCE
Autumn 2014
Quiz 2
1. Here is a list of IQ of 8 randomly selected students. We wish to see if their intelligence is similar to that of the population average of 100. Describe the hypotheses.
Assuming normality, s
Introduction to Statistics Lecture 6
Estimation
October 22, 2015
1
Motivation
Statistical inference is concerned with making probabilistic statements about
unknown quantities.
Examples: means, variances, quantiles and unknown parameters of distributions.
Problem Set - II
November 16, 2013
1. Let Y1 , . . . , Yn denote a random sample from a normal distribution with mean
2
2
(unknown) and variance 2 . For testing H0 : 2 = 0 against H1 : 2 > 0 ,
(a) Device an intuitive (chi-squared) test, and
(b) nd likeli
November 23, 2013
1
Simple linear regression
1.1
Normal simple linear regression
To perform inference we need to make assumptions regarding the distribution of
i.
We often assume they are normally distributed.
The normal error version of the model for sim
Lecture Schedule for W4109 (Fall 2013)
The first half of the course will cover most of Chapters 1-6. The second half of the course will cover most
of Chapters 7-12 from the textbook, with a bit of extra material thrown in (e.g., non-parametric density).
D
1
STATISTICS G6503 STATISTICAL INFERENCE AND TIME SERIES MODELLING
SAMPLE MIDTERM
STUDENTS NAME: _
Directions:
The exam is closed book and closed notes.
You may use a calculator (cell phones & tablets are NOT allowed) and a formula sheet
(both sides of le
Introduction to Statistics Lectures 11 & 12
Estimation
October 19, 2013
1
Motivation
Statistical inference is concerned with making probabilistic statements about
unknown quantities.
Examples : means, variances, quantiles and unknown parameters of distri
Introduction to Probability some problems
September 21, 2013
1
Problems
1. Roll a fair dice repeatedly. Let X be the number of 6s in the rst 10 rolls and
let Y the number of rolls needed to obtain a 3.
(a) Write down the probability mass function of X .
(
Lecture 11
December 7, 2013
1
Tests of Goodness-of-t
In some problems, before we collect data we have some specic distribution in mind for the
data we will observe. We can test if the data indeed comes from the specied distribution
or not.
1.1
The 2 test
StatisticsW4109: Probability and Statistical Inference
Fall 2013
This is a master's / advanced undergraduate level, double-credit introductory
course in probability and mathematical statistics.
Course goals: This course covers basic probability theory and
Practice Problem Set
October 2, 2015
1
Problems
1. A California license plate consists of a sequence of seven symbols: number,
letter, letter, letter, number, number, number, where a letter is any one of 26
letters and a number is one among 0, 1, . . . ,
W4109: PROBABILITY & STATISTICAL INFERENCE
Autumn 2014
Homework 1
Homework is due Thursday, September 18 at 10:10am. You may turn it in in class, at the
beginning of the class, or send it via e-mail to Michael and Diego.
Read Chapter 1 of DeGroot and Sche
Introduction to Probability some problems
September 14, 2013
1
Problems
1. A California license plate consists of a sequence of seven symbols: number,
letter, letter, letter, number, number, number, where a letter is any one of 26
letters and a number is
Method of Maximum likelihood
Suppose random variables X1 , , Xn have joint density or frequency function
f (X1 , , Xn |)
Given observed value Xi = xi , i = 1, , n, the likelihood of is as a
function of x1 , , xn dened as
lk() = f (x1 , , xn |)
We are cons
Refresher on Bayesian and Frequentist Concepts
Bayesians and Frequentists
Models, Assumptions, and Inference
George Casella
Department of Statistics
University of Florida
ACCP 37th Annual Meeting, Philadelphia, PA [1]
Approaches to Statistics
Frequentist
W4109: PROBABILITY & STATISTICAL INFERENCE
Autumn 2014
Homework 4
Homework is due Thursday, November 6 at 10:10am. You need to upload it to courseworks
assignments tab. Read Sections 6.1-6.4 and 7.1-7.8 of DeGroot and Schervish. You may nd
solutions to so
Statistical analysis of neural data:
Monte Carlo techniques for decoding spike trains
Liam Paninski
Department of Statistics and Center for Theoretical Neuroscience
Columbia University
http:/www.stat.columbia.edu/liam
March 31, 2009
1
Contents
1 Often we
Monte Carlo Simulations
October 9, 2012
Motivation
Simulation uses high-speed computer power to substitute for analytical
calculation.
LLN : if we observe a large sample of i.i.d. random variables with finite
mean, then the average of these random varia
Monte Carlo Simulations
October 9, 2012
Motivation
Simulation uses high-speed computer power to substitute for analytical
calculation.
LLN : if we observe a large sample of i.i.d. random variables with finite
mean, then the average of these random varia
Todays outline
Examples from bivariate distributions
Examples of variable transformation
One-to-one transformations
2d transformations
Example: dependent bivariate distribution
21 2
x y for x2 y 1.
4
Notice that range of x depends on y, not independen
Linear regression
Have xi s and yi s which are assumed to be linked through
yi = + xi + i
How do we find and ?
Least squares
Find and such that they minimize
X
(yi yi )2 .
We get
=
P
(x
x)(yi
y)
Pi
(xi
x)2
= y x
MLE
Now assume in addition that i N