Probabilistic Systems Analysis and Applied Probability
ENGINEERIN 6.041

Fall 2010
LECTURE 20
THE CENTRAL LIMIT THEOREM
Usefulness
universal; only means, variances matter
accurate computational shortcut
Readings: Section 5.4
justication of normal models
X1, . . . , Xn i.i.d., n
Probabilistic Systems Analysis and Applied Probability
ENGINEERIN 6.041

Fall 2010
X
X
Estimator
W fW (w)
Readings: Section 9.1
f()
(not responsible for tbased condence
intervals, in pp. 471473)
Estimator ()
p
X X
Estimator
NN
Estimator
p()
cfw_0, 1
X =+W
W fW (w)
Estimator
Probabilistic Systems Analysis and Applied Probability
ENGINEERIN 6.041

Fall 2010
LECTURE 24
Review
Maximum likeliho o d estimation
Have model with unknown parameters:
X pX (x; )
Pick that makes data most likely
Reference: Section 9.3
Course Evaluations (until 12/16)
http:/web
Probabilistic Systems Analysis and Applied Probability
ENGINEERIN 6.041

Fall 2010
p()
p()
p()
N
p()
N
N
p()
Inference models/approaches
pX(x  )
Types of
p()p
Sample Applications()
N
pX(x
Model building versus inferring unknown )
pX(x  )
p
N
p()
p() ()
Readings: Sections
Probabilistic Systems Analysis and Applied Probability
ENGINEERIN 6.041

Fall 2010
LECTURE 18
Review
Markov Processes III
Assume a single class of recurrent states,
aperiodic;
aperiodic. Then,
plus transient states. Then,
LECTURE 20
Review
Assume a single class of recurrent states