18.443. Practice test 1.
Consider the family of distributions with p.d.f.
f (x| ) = x1 , for 0 < x < 1, and > 0.
Consider an i.i.d. sample X1 , . . . , Xn from this distribution. As always, the
underlying parameter for this sample is unknown. In probl
Testing hypotheses about parameters
of normal distribution.
T-tests and F-tests.
We will postpone a more systematic approach to hypotheses testing until the following
lectures and in this lecture we will describe in an ad hoc way T-tests and F-t
Simple linear regression.
Let us look at the cigarette dataset from  (available to download from journals website)
and . The cigarette dataset contains measurements of tar, nicotine, weight and carbon
monoxide (CO) content for 25 brands o
Chi-squared goodness-of-t test.
Example. Let us start with a Matlab example. Let us generate a vector X of 100 i.i.d.
uniform random variables on [0, 1] :
Parameters (100, 1) here mean that we generate a 1001 matrix or uniform ra
Tests of independence and
In this lecture we will consider a situation when our observations are classied by two dierent
features and we would like to test if these features are independent. For example, we can ask
if the number of
Multiple linear regression.
Let us consider a model
Yi = 1 Xi1 + . . . + p Xip + i
where random noise variables 1 , . . . , n are i.i.d. N(0, 2 ). We can write this in a matrix
Y = X + ,
where Y and are n 1 vectors, is p 1 vector and X is
Suppose that we have an i.i.d. sample X1 , . . . , Xn with some unknown distribution P and we
would like to test the hypothesis that P is equal to a particular distribution P0 , i.e. decide
between the following hypothe
Gamma distribution, 2-distribution,
Fisher F -distribution.
Gamma distribution. Let us take two parameters > 0 and > 0. Gamma function
() is dened by
x1 ex dx.
If we divide both sides by () we get
1 1 x
x e dx
Maximum Likelihood Estimators.
Matlab example. As a motivation, let us look at one Matlab example. Let us generate
a random sample of size 100 from beta distribution Beta(5, 2). We will learn the denition
of beta distribution later, at this poin
Multivariate normal distribution and
We start with several simple observations. If X = (x1 , . . . , xk )T is a k 1 random vector
then its expectation is
EX = (Ex1 , . . . , Exk )T
and its covariance matrix is
Cov(X) = E(X EX)(
Properties of MLE: consistency,
In this section we will try to understand why MLEs are good.
Let us recall two facts from probability that we be used often throughout this course.
Law of Large Numbers (
Testing simple hypotheses. Bayes
Let us consider an i.i.d. sample X1 , . . . , Xn X with unknown distribution P on X . Suppose
that the distribution P belongs to a set of k specied distributions, P cfw_P1 , . . . , Pk . Then,
Condence intervals for parameters of
Let us consider a Matlab example based on the dataset of body temperature measurements
of 130 individuals from the article . The dataset can be downloaded from the journals
18.443. Practice test 2.
(1) Given a sample 5, 1, 4, 1, 2, 3 from Poisson distribution (), construct
the most p owerful test for
H0 : = 1 vs. H1 : = 2,
with level of signicance = 0.05. Test H0 .
(2) p. 561, no. 1.
(3) p. 574, no. 4.
(4) Suppose that in th
Goodness-of-t for composite
Example. Let us consider a Matlab example. Let us generate 50 observations from N(1, 2):
Then, running a chi-squared goodness-of-t test chi2gof
H = 0,