Math 4056: Notes on simple hypothesis testing
Lecturer: Dr. A. Ganguly
Essential facts:
Suppose that X1 , X2 , . . . , Xn are iid with common density function (can also be pmf) f (x, ), and
using the data you want to test the following hypothesis:
H0 : 0
Math 4056: Practice problem set - 1
Lecturer: A. Ganguly
August 25, 2016
Explain all the necessary steps in your answers.
1. In a population the probability of a person getting infected by a disease of type I is 0.65, and
the probability of a person getti
Math 4056: Practice problem set - 2
Lecturer: A. Ganguly
August 30, 2016
Explain all the necessary steps in your answers.
For each of the following distributions, write down its p.m.f/density. Then calculate its expectation
and variance.
(a) X Binomial(n,
M4056 Quiz 6, Oct. 1, 2010
Name
Recall that the Poisson distribution with parameter > 0 is the discrete distribution with
pmf
x
f (x|) = e
, x = 0, 1, 2, . . .
x!
Suppose a sample X = (X1 , . . . , Xn ) is drawn from a Poisson distribution with unknown
n
M4056 Quiz 5, Sept. 24, 2010
Name
1. Recall that the Poisson distribution is a discrete distribution with pmf
f (x|) = e
x
,
x!
x = 0, 1, 2, . . .
a. Let X = (X1 , . . . , Xn ) be a sample from a Poisson distribution. Write the sample
pmf explicitly as a
M4056 Quiz 4, Sept. 17, 2010
Name
Suppose
U =X
X 2 Y and V = X +
X2 Y .
Then
X = 1 (U + V ) and Y = U V
2
(as you can easily verify). Now suppose these are all random variables, and suppose that
the joint distribution function of X and Y is fX,Y (x, y ).
M4056 Quiz 3, Sept. 10, 2010
Name
1) Complete the statement in three dierent ways, in one, referring to the pdf s of X and
Y , in the next to the cdf s and in the third referring to the probabilities that X and Y are
in specied sets:
Random variables X an
M4056 Quiz 2, Sept. 3, 2010
Name
1) Fill in the blanks:
i) E (aX + bY ) =
ii) E (X 2 ) =
iii) Var (aX + b) =
iv ) If X and Y are independent, Var (aX + bY ) =
2) Suppose X has moment generating function MX (t) = E (etX ). Let a and b be constants.
Compute
M4056 Home Work Answers
December 1, 2010
Problem 12 (recalled)
In this problem, we are given f (x | ) = ex . If X = (X1 , . . . , Xn ) is an i.i.d. sample,
n
then f (x | ) = n e(x1 +xn ) = ex = f (x | ).
d
The log likelihood function is ( ) = n(ln x ). Si
Math 4056: Practice problem set - 3
Lecturer: A. Ganguly
September 8, 2016
By convention, all the probability density functions used here are assumed to be equal to zero
wherever their values are not explicitly specified.
1. Suppose X N (2, 9). Find an ex
Time: 50 min, Total points: 100
Name:
Math 4056: Sample Exam 1
Lecturer: A. Ganguly
September 17, 2016
Explain all the necessary steps in your answers.
1. Suppose that X N ( = 2, 2 = 9).
(a) Find P (2 X < 5) in terms of , the cdf of the standard normal di
Math 4056: Practice problem set 11
Problems on simple hypothesis testing
Lecturer: Dr. A. Ganguly
i.i.d
1. Suppose that X1 , X2 , . . . Xn Poisson(). Suppose you want to test the hypothesis:
H0 : = 1
vs H1 : = 2.
(a) Find the general form of the critical
Math 4056: Practice problem set 10
Problems on sufficiency
Lecturer: A. Ganguly
Nov 2, 2016
Explain all the necessary steps in your answers.
i.i.d
1. Suppose that X1 , X2 , X3 , . . . , Xn Poisson(). Find a sufficient statistic, T , for . Verify
that the
Math 4056: Practice problem set 13
Problems on regression
Lecturer: Dr. A. Ganguly
1. Assume that you are given the data points (X1 , Y1 ), (X2 , Y2 ), (X3 , Y3 ), . . . , (Xn , Yn ). Consider
the linear regression model:
Yi = a + bXi + ei ,
where the ei
Time: 50 min, Total points: 100
Name:
Math 4056: Sample Exam 3
Lecturer: A. Ganguly
November 19, 2016
Explain all the necessary steps in your answers.
i.i.d
1. Suppose that X1 , X2 , X3 , . . . , Xn Binomial(N, ), where N is known.
(a) Find a sufficient s
Math 4056: Practice problem set 12
Problems on composite hypothesis testing
Lecturer: Dr. A. Ganguly
i.i.d
1. Suppose that X1 , X2 , . . . Xn Normal(, 2 ). Suppose you want to test the hypothesis:
H0 : = 0
vs H1 : 6= 0 ,
for some known number 0 .
Describe
Math 4056: Practice problem set - 4
Lecturer: A. Ganguly
September 8, 2016
By convention, all probability density functions are assumed to be equal to zero at points where
their values are not specified. All probability distribution functions are assumed
Math 4056: Practice problem set 9
Problems on point estimation, confidence intervals, efficiency
Lecturer: A. Ganguly
October 6, 2016
Explain all the necessary steps in your answers.
1. (a) Find the information I() for the Poisson() distribution. Recall t
Math 4056: Practice problem set - 6
Lecturer: A. Ganguly
September 15, 2016
Explain all the necessary steps in your answers.
1. Let X Geometric(p). Verify that its m.g.f is given by MX (t) =
pet
1(1p)et .
Solution: Recall that X takes values in the set cf
Math 4056: Practice problem set - 5
Lecturer: A. Ganguly
September 13, 2016
Explain all the necessary steps in your answers.
1. Suppose that X and Y are independent Exponential() random variables. Find the joint
density of X + Y and X/Y . Are they indepen
Math 4056: Practice problem set - 7
Lecturer: A. Ganguly
September 22, 2016
Explain all the necessary steps in your answers.
1. Suppose X = (X1 , X2 , X3 )T has a multivariate normal distribution with mean vector =
(0, 2, 1)T and covariance matrix
1 0 0
Time: 50 min, Total points: 100
Name:
Math 4056: Sample Exam 2
Lecturer: A. Ganguly
October 15, 2016
Explain all the necessary steps in your answers.
i.i.d
1. (25 points) Suppose that X1 , X2 , X3 , . . . , Xn N (, 2 ). Prove that X and S 2 are independen
M4056 Analysis of Variance, I
November 22-24, 2010
1. Introduction and Goal
Let X be a normal random variable with mean X and variance 2 . Let Y be another
normal random variable with mean Y and the same variance 2 as X . In the lectures
of November 17 an
M4056 Condence Intervals
November 15, 2010
Let be a parameter of a probability distribution. A condence interval for is a random
interval, calculated from a sample, that contains with some specied probability. (A
random interval is an interval [L(X ), U (
M4056 Lecture Notes.
September 10, 2010
Transformations
These notes are intended to clarify a point in the last lecture.
Suppose we have two copies of Rn , one with coordinates x1 , . . . , xn and one with coordinates u1 , . . . , un . Suppose also that w
M4056 Lecture Notes.
September 8, 2010
Sampling from a normal distribution II
Before getting to the meat of this lecture, we provide some orientation. Bear in mind the
kinds of inferential tasks we might face:
i) We are sampling from a population known to