University of British Columbia, Department of Statistics
STAT 305, Introduction to Statistical Inference, 2014/15 Term 2
Instructor: William J. Welch
Course web page: http:/ugrad.stat.ubc.ca/~stat305
Overview
An introduction to likelihood and Bayesian pri
Worksheet 5.1 Clicker Question 1
Today, STAT 305 makes some connections with STAT 306 .
Linear models or regression models like
(
)
Yi N i , 2 , where i = xi
are covered in STAT 306, for example.
Have you seen linear models before?
A Yes, I have taken STA
Worksheet 2.2 Clicker Question 1
(
)
Let Y1 , . . . , Yn be independent N , 2 random variables, let Z be a
standard normal random variable, and define the value z0.975 by
Pr(Z < z0.975 ) = 0.975. We agreed that
Pr(Y z0.975 < < Y + z0.975 ) = 0.95
n
n
is a
Worksheet 1.1 Clicker Question 1
E(Y ) is found here by computing:
A E(Y ) = Y
B E(Y ) = y
C E(Y ) =
D E(Y ) =
1 2
E E(Y ) = y 1 e 2 y dy .
2
c Copyright William J. Welch 2017. All rights reserved.
Not to be copied, used, or revised without explicit wri
Worksheet 3.2 Clicker Question 1
Variance is defined as
Var(Y ) = E(Y E(Y )2 = E(Y 2 ) (E(Y )2 .
Variance is a measure of the variability of what?
A A value of a random variable
B A random variable
C A parameter with a Greek letter like 2
D A number like
Worksheet 1.3 Clicker Question 1
For one MA line, write for the probability a mutation occurs at a specific
bp. Furthermore, assume that whether or not mutations occur is
statistically independent across the n bp.
Let Y be the total number of mutations fo
STAT 305
2016/2017 Term 2
THE UNIVERSITY OF BRITISH COLUMBIA
DEPARTMENT OF STATISTICS
STAT 305 Introduction to Statistical Inference
Quiz 1 Solution
1
999 = 0.2707
1. (a) Pr(X = 1) = nx x (1 )nx = 1000
1 (0.002) (0.998)
y
2 1
(b) Pr(Y = 1) = e y! = e 1!2
STAT 305, Quiz2
2016/2017 Term 2
1. (a) The log likelihood is
ln(fY1 ,.,Yn (y1 , ., yn |) =
n
X
!
yi n
ln(1 ) + n ln()
i=1
To find a turning point, set
Pn
yi n n
ln fY1 ,.,Yn (y1 , ., yn |) = i=1
+ = 0,
1
P
which has the solution
= n/ ni=1 yi = 1/
y .
Worksheet 8.1 Clicker Question 1
Suppose a CEO is born in June or July with probability . What is a
reasonable probability model for Y , the number of CEOs out of n = 375
born in June or July?
A Y Pois ()
B Y Pois (n)
C Y N (n, n(1 )
D Y Geom0 ()
E Y Bin
Worksheet 7.6 Clicker Question 1
What values of the T would be evidence against H0 in favour of Ha ?
A Negative values of large enough magnitude
B Positive values of large enough magnitude
C Negative or positive values of large enough magnitude
D Values c
Worksheet 4.2 Clicker Question 1
.
The binomial PMF is
( )
n y
fY (y | ) =
(1)ny
y
(y = 0, 1, . . . , n; n = 1, 2, . . . ; 0 < < 1).
As a function of for fixed y the function is called the binomial likelihood.
How do we interpret the likelihood function?
Worksheet 2.1 Clicker Question 1
(
)
Y N , 2 . What is the MGF of
Z=
Y
?
A et
1 2
B e 2t
C e at MY (bt)
1
D e t+ 2
2t2
E MZ (t).
c Copyright William J. Welch 2017. All rights reserved.
Not to be copied, used, or revised without explicit written permissi
CLASS SCHEDULE 2015/03/20
Further revisions / additions are unlikely; the quiz dates are fixed.
Lab TOPIC
Worksheet Course notes
RICE (3rd edition)
Mon Jan
5
Introduction; Quiz #0 diagnostic of MATH/STAT 302 prerequisite
Wed Jan
7
Moment generating functi
STAT 305
Introduction to Statistical Inference
Known Typos in STAT 305 Course Notes
Thanks to my colleague Professor John Petkau for his careful reading to compile this list of typos. Most
are quite minor so you have probably already determined what was i
STAT 305: Introduction to Statistical Inference
Asymptotic Properties of Maximum Likelihood Estimators
March 14, 2014
Context: We limit attention to the special case of a likelihood that is based
on independent and identically distributed random variables
Worksheet 3.1 Clicker Question 1
Let Y be a random variable with a Bin (n, ) distribution. If n is large and
is small, the PMF of Y can be approximated by a Pois ( = n)
distribution.
What is the justification?
A With large n, the central limit theorem ca
Worksheet 1.2 Clicker Question 1
The exponential distribution has PDF
fY (y ) = e y
(0 < y < ; > 0).
What is the interpretation of, say, fY (1) = e ?
A fY (1) is the probability that Y takes the value 1
B fY (1) is the probability that Y < 1
C fY (1) is t
Worksheet 2.3 Clicker Question 1
Let be the proportion in the population who agree with the statement
A changing climate presents a significant threat to our economic future.
In a random sample of size n = 1000, let X be the number of people
agreeing with
Worksheet 5.2 Clicker Question 1
Recall in Chapter 2 on random samples from the normal distribution that
we defined the sample variance as
1
(Yi Y )2 .
n1
n
S2 =
i=1
Why is there a divisor of n 1?
A There are n 1 degrees of freedom in the Yi Y
B It leads
Worksheet 4.1 Clicker Question 1
Consider either the treated or control data, where we have y bees
returning out of a sample size of n. What is a reasonable probability model
for a random variable Y that gave the value y ?
A Normal with mean and variance
Worksheet 4.3 Clicker Question 1
The joint PMF of a random sample of n IID Geom0 () random variables is
fY1 ,.,Yn (y1 , . . . , yn | ) =
n
n
Pr(Yi = yi | ) = (1 )
i=1 yi
n .
i=1
Why?
A It is a property of the geometric distribution
B The geometric distrib
STAT 305: Wrap Up
Will Welch (Instructor)
JanuaryApril 2015
Will Welch (Intro Stat Inference)
STAT 305
JanuaryApril 2015
1/8
Thinking Like a Statistical Expert
Put numbers in formulas . . .
Principles of statistical modelling and methodology to tackle a w