MA 2B ASSIGNMENT 1 SOLUTIONS
KC BORDER
Exercise 1. (10 pts) Go to the U.S. Centers for Disease Control, National Vital Statistics Report, vol. 61, no.3 (Sep. 24, 2012), http:/www.cdc.gov/nchs/data/nvsr/nvsr61/nvsr61_03.pdf,
and nd the appropriate life tab
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 21:
1
KC Border
Winter 2013
Testing categorical data
Review of the multinomial distribution
The multinomial distribution generalizes the binomial distribution to r
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 17:
1
KC Border
Winter 2013
Estimation, continued
The method of moments
Let X1 , . . . , Xn be independent and identically distributed with density f (x; 1 , . . .
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 16:
1
KC Border
Winter 2013
Estimation
Get to know the likelihood function
Let X1 , X2 , . . . , Xn be independent and identically distributed with common pdf
f (x
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 15:
KC Border
Winter 2013
What is Statistical Inference?
1
Objects of statistics
1.
Estimation.
2.
Hypothesis testing.
3.
Prediction.
2
Parametric Distributions in
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
KC Border
Winter 2013
Lecture 14: Conditional Distribution and
Conditional Expectation
1
Conditioning on a Random Variable: The discrete case
Let X and Y be discrete rando
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 13:
1
KC Border
Winter 2013
Joint Distributions and the
Bivariate Normal
Discrete Joint Distributions
For X and Y discrete random variables on a probability space
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 10:
1
KC Border
Winter 2013
Order Statistics; Hazards
Working with densities
Recall that a distribution has density if there is a function f 0 such that the
cdf F
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
KC Border
Winter 2013
Lecture 8: Joint Distributions:
Non-independence
1
Where have we used independence?
So far we have used independence to construct probability measure
MA2B
SOLUTIONS TO HOMEWORK 4
INSTRUCTOR:
KIM BORDER
205 BAXTER HALL
WINTER 2013
4218
Exercise 1
Essentially all the code was given. In the listing on the next page you can nd a
complete R script, which produces desired graphics for Binomial64 and Binomial
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 7:
1
KC Border
Winter 2013
The Law of Large Numbers
Preliminary Inequalities
Markovs Inequality bounds the probability of large values of a nonnegative random vari
1
Should you regress Y on X or vice-versa?
The answer to that question is not a statistical question, it is a scientic
one. Do you have a theory that makes one variable dependent, and the other
independent? The statistical question is what dierence does i
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
KC Border
February 2013
Dierentiating under an integral sign
In the derivation of Maximum Likelihood Estimators, or the CramrRao Lower Bound,
we dierentiated under an inte
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 18:
1
KC Border
Winter 2013
Estimation, one more time
Estimating functions of parameters
Suppose I dont care about per se but some function g(). E.g., suppose I wa
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 9:
1
KC Border
Winter 2013
The Normal Distribution and the
Central Limit Theorem
When are distributions close?
We now come to the topic of when distributions are c
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 12:
1
KC Border
Winter 2013
Get to Know Your Distributions
Bernoulli
Bernoulli(p)
The Bernoulli distribution is a discrete distribution that generalizes coin tossi
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 27:
1
KC Border
Winter 2013
Miscellany II
A big mea culpa
A question in lecture the other day made me realize there are some notation
discrepancies between some of
CALIFORNIA INSTITUTE OF TECHNOLOGY
Department of Mathematics
The simple random walk and the law of averages
KC Border
March 2013
v. 2013.03.27:16.16
In 1950 William Feller published An Introduction to Probability Theory and Its
Applications [2]. According
MA 2B ASSIGNMENT 3 SOLUTIONS
KEVIN LINGHU
Exercise 1 (Problem 3.3.15 in Pitman). (15 pts)
a. Let X and Y be independent random variables. Show that
Var(X Y ) = Var(X + Y ).
b. Let D1 and D2 represent two draws at random with replacement from a population
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 11:
1
KC Border
Winter 2013
The Poisson Process
The Exponential is Memoryless
The exponential distribution is memoryless in that
)
(
P T > t + s T > t = P (T > s)
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 19:
1
KC Border
Winter 2013
Test statistics for the Normal
Testing hypotheses based on normality
Larsen
For a sample X1 , . . . , Xn of independent and identically
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 26:
1
KC Border
Winter 2013
Miscellany
Prediction intervals in the linear model
Recall:
Y = X 1 1 + + X K K +
(1)
OLS = (X X)1 X y
(2)
y = X OLS
e = y y = y X OLS
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 20:
KC Border
Winter 2013
Test statistics for the Normal,
continued
[Review t-testing; one-sided vs two sided; output of software]
1
Dierence of means, again
Given
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 24:
1
KC Border
Winter 2013
The Standard Linear Model:
Hypothesis Testing
Condence intervals
From last time:
Let there be T observations on K regressors X (includi
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 25:
1
KC Border
Winter 2013
Analysis of Variance
ANOVA is Multiple Regression in disguise
According to Wikipedia, the law of the instrument was formulated by Abrah
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 22:
1
KC Border
Winter 2013
The Multivariate Normal and the
Chi-square Test
The Normal density
Recall that the Normal N (, 2 ) has a density of the form
1
1
1
e 2
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 6:
1
KC Border
Winter 2013
Higher expectations
Bonus Question
Since a long weekend is coming up, I am giving you a bonus question. The rst
two answers that I recei
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 1:
1
KC Border
Winter 2013
Introduction
Uncertainty, randomness, and probability
Karl Ors O Fortuna is a musical tribute to Fortune. The lyrics are from an irrever
CALIFORNIA INSTITUTE OF TECHNOLOGY
Ma 2b
Introduction to Probability and Statistics
Lecture 8:
KC Border
Winter 2014
The Law of Large Numbers
Relevant textbook passages:
Pitman [7]: Section 3.3
Larsen & Marx [6]: Section 4.3
8.1
DeMoivreLaplace Limit Theo
HOMEWORK 8 SOLUTIONS
1. Problem 2 (Continued)
6, Despite having nearly identical statistical properties, the scatter plots display very
disparate relationships between X and Y. This indicates the importance of using graphing
techniques to visualize data.