Stat 150 Midterm 1 Spring 2015
Instructor: Allan Sly
Name:
SID:
There are 4 questions worth a total of 48 points. Attempt all questions and show your
working - solutions without explanation will not receive full credit. Answer the questions in
the space p
Worksheet # 1
Statistics 150, Pitman, Spring 2013
Review of basic probability framework. Limits of random variables. Reading: All of Chapter
1. Focus: 1.5.5 MGFs and PGFs (z-transforms), 1.6.3 Cherno bounds, 1.7.5/6 Convergence
of RVs. Monotone Convergenc
19 April 2013
Last Couple Weeks.
Gaussian Processes and Brownian Motion (Ch. 3)
Martingales (Ch. 9)
Brownian Motion.
Motivation.
Understand scaling limits of random variables
Direct modeling of stock market, etc.
(Bt , t 0) will be a continuous time a
Worksheet # 5
Statistics 150, Pitman, Spring 2013
Topics: First Return Probabilities for Simple Random Walk , Occupation Times for Symmetric Simple Random Walk , Recurrence/Transience for Simple Random Walk , Stopping
Times and Walds Identity
Reading: Top
17 April 2013
Today.
Complete proof of F-K formula
Discuss issues around transforms
Prepare for discussion of continuous time / space processes
If you want to get ahead, read the Gaussian section, prepare for Brownian motion
F-K Formula. From last time, w
15 April 2013
Feynman-Kac Formula.
The problem. Let (Xt , t 0) be a continuous time Markov chain with state space S B ,
where
S is a set of internal states.
B is a nonempty set of boundary (absorbing) states.
Assume it is possible to reach B from any in
8 March 2013
Today / Announcements.
Exercise
Wrap up discussion of branching processes and random walks
Midterms: 2010 MT w/ Solns, 2009 MT w/o Solns, 2006 MT w/o Solns
Exercise. Plot the probability generating function for a geometric random variable
20 March 2013
Today. Continue continuous time Markov Chains
Poisson Process on the Line
Connections to Renewal Theory
Poisson Process on the Line. Consider a process on the line with rate 1 per minute. Let
Tn := X1 + + Xn . Maybe its a really busy bus s
22 March 2013
Reading.
Poisson Process
Renewal Theory (Limit Theorems)
Pitmans notes:
Poisson Process
Jump Hold Description
Transition Semigroups
Limit Distribution of Continuous Time Markov Chains
Poissonization of Discrete Time Markov Chains
Ge
1 April 2013
Brief Review of Poisson Methods / Tools / Tricks (Must Know).
Let us interpret this picture. Remember:
1
U4
U1
U5
U3
p
U2
U6
0
T1
T2 T3
T4
T5
T6
P P ( )
=+
This is the important picture for this lecture
T0 = 0 < T1 < T2 <
(Ti Ti1 , i 1) a
3 April 2013
Brief Review of Last Class. Any continuous time Markov chain (X (t), t 0) with nite
state space S can be constructed as X (t) = YN (t) , where N is a Poisson Process with
parameter , and Y is a jumping chain with transition matrix in discrete
5 April 2013
Last time, we developed insight into structure of a two-state chain in continuous time.
0
1
The insight:
0
1
+
1) The steady state equilibrium is given by:
+
2) If you run a PP( + ), you can thin to get the transitions, i.e. the types of arr
8 April 2013
Wrap Up Ehrenfest Urn. Recall: in this model, we have N labeled balls, two urns labeled
I and II . Let Xt be the number of balls in urn I at time t. Each particle ips boxes at rate
1, i.e. each particle has its own Poisson process of rate 1.
10 April, 2013
Queuing Models. General idea: customers arrive at a service station expecting a service.
They queue up for service, receive the service, then depart. The classic classication of
queues follows the / / format. Typically in the rst dot goes M
12 April 2013
Laplace Transforms. Consider a random variable X 0. Its Laplace transform is () :=
X () = E[eX ].
Motivation. The Laplace Transform is a kind of moment generating function. Recall that
a moment generating function is E[eX ], for some real (o
6 March 2013
Today.
Practice Problem
Branching Processes and Random Walks
Practice Problem. Sketch the binomial(n, 1 ) generating function for n = 1, 2, 3, 4. An2
swer:
1/2
1/4
1/8
1/16
To see this, notice that the general form of the generating functio
248 Thursday February 7 1011 Evans Victor Panaretos Lausanne
Spectral Analysis of Functional Data
Functional data analysis deals with statistical problems where each
observation in a sample is an entire realization of a random function,
the aim being to i
Point processes
definitions, displays, examples, representations,
algebra, linear quantities
Point process data
points along the line
radioactive emissions, nerve cell firings, wildfires, accidents,
Describe by: a) 0 1 < 2 < . < N < T in [0,T)
b) N(t) = #
Stat 150 Homework # 8 Solutions
1: If the density function of a distribution has the form ck, xk1 ex , then ck, must be the
normalizing constant xk11 x dx and immediately from the functional form of the density
e
(i.e. being some constant times xk1 ex ) w
Stat 150 Spring 2015 Syllabus
Available online at http:/www.stat.berkeley.edu/~sly/Stat150Spring2015Syllabus.pdf
Instructor: Allan Sly
GSI: Jonathan Hermon
Course Webpage: http:/www.stat.berkeley.edu/~sly/STAT150.html
Class Time: MWF 12:00 - 1:00 PM in ro
Stat 150 Homework # 6 Solutions
1: Denote t0 = 0 = x0 , si := ti ti1 and yi = xi xi1 . Then 3 cfw_N (ti ) = xi =
i=1
3
i=1 cfw_N ([ti ti1 ) = yi (up to an event of 0 probability in which there was an arrival in
one of the times cfw_t1 , t2 , t3 . Thus b
Stat 150 Homework # 9 Solutions
1: By the stationarity of BM this is like 2 + W1 given that W2 = 2.
1
First way - using scaling: Bt = 2 W2t is a standard BM if Wt is. So what we have is like
=
2 + 2B1/2 given that B1 = 2. You saw in lecture that given tha
Stat 150 Homework # 4 Solutions
1: Let X Geometric(p). Then
p[(1 p)s]k =
GX (s) =
k=0
p
,
1 (1 p)s
Let Y1 , Y2 , . . . be i.i.d. Bernoulli(q) random variables, such that X, Y1 , Y2 , . . . , are independent. Dene Y := X Yi . Then Y Bin(X, q). We have that
Stat 150 Homework # 7 Solutions
A summary of some useful facts:
A continuous-time Markov chain (Yt )tR+ on a finite state space can be described in several
ways:
(i) By its generator G. In which case Ht (x, y) := Pr[Yt = y | Y0 = x] = (etQ )x,y . In which
NY Times 25 November 2008
Stat 153 - 24 Nov 2008 D. R. Brillinger
Chapter 14 - Examples continued
Question
Data
Analyses
Conclusions
Why? Chatfield's example 14.1
Monthly mean air temperature at Recife 1953-1962
Table 14.1 - doesn't indicate source ?
Chat
Point processes on the line. Nerve firing.
Stochastic point process. Building blocks
Process on R cfw_N(t), t in R, with consistent set of
distributions
Prcfw_N(I1)=k1 ,., N(In)=kn
k1 ,.,kn integers 0
I's Borel sets of R.
Consistentency example. If I1 ,
Stochastic models - time series.
Random process.
an infinite collection of consistent distributions
probabilities exist
Random function.
family of random variables, cfw_Y(t;), t Z,
Z = cfw_0,1,2,., a sample space
Specified if given
F(y1,.,yn;t1 ,.,tn ) =
dynamic spectrum
Operations/systems.
Processes so far considered:
Y(t), 0<t<T
using Dirac deltas includes
Y(x,y), 0<x<X, 0<y<Y
includes
Y(x,y,t), 0<x<X, 0<y<Y, 0<t<T
includes
and more
time series
point process
image
spatial point process
spatial-temporal
Statistics 248
D.R. Brillinger
Process data
indexed values Y(), : time, space, set, function
curves, surfaces, shapes, measures, images, sequences, scatter,
spirals, trajectories, mosaics - and they may be moving
Random process data
Put process datums in