Transcript Review and Response - people.stat.sfu.ca

Review for Midterm
Including response to student’s
questions Feb 26.
Sampling Framework
• Population of Interest
Sometimes know dist’n model (shape)
Usually don’t know parameters
• Sample
A set of n numbers where n is the sample
size - usually drawn at random from the
population (like “tickets from a hat”).
Random Sampling may be with or without replacement
i.e.
SWR vs SWOR
Inference Goal
• To use sample data to estimate
population parameters.
• Example: use X to estimate 
• But, would like accuracy of estimate.
X
• If unbiased,
accuracy
is
just
SD
of
,

estimated by s
n
Sampling Distribution of
X
• Approx Normal (CLT)
• Expected Value of X is  (the
population mean)
• SD of is X is / n called standard error
( is the population SD and n is the
sample
size)
• Usually,
  and  must be estimated
 from the sample, using X and s.

Conditional Probability
• P(A|B) = P(A and B)/P(B) where A and B are
events (i.e. sets of sample space outcomes)
•Example: Urn [3 Red and 5 Green] SWOR
•Let R1 be event that the first draw is red
•Let G2 be event that the second draw is green
P(R1|G2) = ?
=P(G2 | R1) * P(R1) / P(G2)
=(5/7)*(3/8) / P(G2)
P(G2) = P(G2 and R1) + P(G2 and R1’)
= P(G2| R1)P(R1) + P(G2| R1’)P(R1’)
= 5/7 * 3/8 + 4/7 * 5/8 = 5/8
So P(R1|G2) = 3/7
Uniform Distributions
• Discrete P(X=x) = 1/n
Mean = (n+1)/2 SD =
• Continuous
x=1,2,3,…,n
(n 2 1) /12
f X (x) 1/c

for 0<x<c
and 0 otherwise.
Mean = c/2 SD =
c
12
Model Links
•
•
•
•
•
•
•
Waiting time for kth success - neg. bin.
Waiting time for rth event - gamma
Waiting time for first success - geom.
Waiting time for first event - exponential
Number of events during time - Poisson
---------------Time between successive events - exp
Shape of Gamma family
•
•
•
•
•
•
•
Parameters , 
 = 1 -> exponential
 large -> normal
 moderate -> right skew
 contracts or expands scale.
1/ 2

Mean =  SD =

Determining reasonable , 
(Use Mean&SD)

The bootstrap - bare bones
• A statistic t(x1,x2,…,xn) estimates
parameter 
• Need: SD of t(), since it is precision of
estimate.
• Method: Re-Sample (x1,x2,…,xn) many
times and compute t() each resample.
Then compute SD of resample values of
t().
• Result - an estimate of the precision of
t() as an estimate of .
Overview of Ch 1-6
Ch 1 - “Distribution” - tables and graphs
Ch 2 - Probability Calculus - counting rules, conditioning
Ch 3&4 - Models and Connections
Ch 5 - CLT and sampling distribution of a statistic
Ch 6 - Estimators, Estimates, and the Bootstrap