Confidence Intervals for a Parameter
February 2, 2014
Confidence Intervals (CI)
Point estimators are not right( at least when the statistical model
is continuous). Why?
Can we find an interval which contains the parameter with some
certainty?
A confidence
Power and Sample Size
The power of a test of hypothesis with xed signicance level is the
probability that the test will reject the null hypothesis when the alterna>ve is
true.
In other words, power is the proba
Sampling Distributions
January 7, 2014
Random Sample
I.I.D. random variables: If X1 , . . . , Xn are the outcomes of a
series of random experiments which are identical and have nothing
to do with each other, then they are independent and identically
distr
Review
January 5, 2014
Methods of Distribution Functions and Transformations
How do we find the PDF of a random variable Y = v (X )?
Write the CDF of Y : FY (y ) = P(v (X ) y )
Invert the inequality
Differentiate the CDF to get the PDF of Y
If v(X) is
MIDTERM
Name (Last name ﬁrst)
Student ID 7%:
$5 [wHQ Ar
Show all your work and answers in the space
provided, in ink. Pencils may be used, but
then your exams cannot be regraded if grading
issues arise.
1 2 3 4
17 18 15 10
Results
Total 1.[l7pts] Let
Method of Moments (MoM)
Scenario: Let X1 , X2 , . . . , Xn be a random sample from a
population with distribution fX (x; ), with = (1 , 2 , . . . , k ). We
would like to estimate
Idea: Use the mean observed data to estimate the expected value
of the dist
Hypothesis Testing
March 9, 2014
P-value
Coin tossing example: A coin is tossed 500 times. It lands heads
275 times. Find the p-value of the hypothesis test
H0 : p = 1/2 vs. Ha : p 6= 1/2
What is the chance of observing something like what we observed
if
Hypothesis Testing
March 17, 2014
Neyman-Pearson Lemma
Finding best tests (also called most powerful tests): Find a
rejection region (RR) such that
1. The probability of RR under H0 is less than some significant
level
2. The probability of RR under Ha is
Estimation (continued)
January 20, 2014
Consistency Result
Recall: An estimator n is consistent for if if converges in
P
probability to . Notation: n
Theorem An unbiased estimator n for is a consistent estimator
of if
lim V (n ) = 0
n
Proof: According to
WELL KNOWN DISTRIBUTIONS
DISCRETE DISTRIBUTIONS
Bernoulli(p)
Probability mass function:
(
fX (x) = Pr[X = x] =
Mean and Variance:
px (1 p)1x x = 0, 1; 0 p 1
0
elsewhere
E[X] = p
V ar[X] = p(1 p)
Moment Generating Function: MX (t) = (1 p) + pet
The Bernou
Estimation
January 13, 2014
Statistical Inference
The process of statistical inference is the opposite of the process
followed in the theory of probability
In the theory of probability, a probability model (a probability
distribution, that is) is given an
Sampling Distributions Continued
January 11, 2014
Normal Distribution
Recall
If X Normal(, 2 ) then
1
2
exp 2 (x ) I(,) (x)
fX (x; , ) =
2
2 2
E [X ] =
1
Var [X ] = 2
The MGF of X is
2t 2
MX (t) = exp t +
2
Normal Distribution
Result Let X1 , X2 , . . .
Question 1
Correct
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Question text
A system in which relatively low-skilled workers use specialized machinery to produce
high volumes of standardized goods describes _.
Select one:
a. Craft production
b. Lean production
c
University of Toronto Department of Statistics STA261H Probability and Statistics II Spring 2011
Lectures:
Mondays 3:10 5 p.m. and Wednesdays 3:10 4 p.m. in MB128 (L0101) Wednesdays 7:10 10 p.m. in SS1069 (L5101) Instructor: Dr. Hadas Moshonov E-mail: had