#PSTAT 160 B
#Dingming Li 8992059
#Hw1 Python 3
import numpy as np
import matplotlib.pyplot as plt
#problem a)
def maxExponentialRV():
a=np.random.exponential(1,3)
b=np.amax(a)
print a,b
#problem b)
#1)
n=0
for i in range (1000):
a=np.random.exponential(1
Announcements
1 / 22
Homework 2 due tonight
Review: Poisson process, Poisson Point process, Event times,
Conditional probability, Simulation
Today:
I
Generalization of Poisson process,
I
I
I
Non homogeneous Poisson process
Compound Poisson process
Mixed
Chapter 6. Continuous time Markov Chain
1 / 22
Continuous time Markov Chain
We say the stochastic process fX (t ); t 0g is a
continuous-time Markov chain with states N0 ,if for every
s ; t 0 , i ; j 2 N0
P(X (t
+
s)
=
j jfX (s )
=
i g \ F (s
) =
P(X (t
+
1 / 18
Recall Poisson Process
A counting process X () is called a Poisson process with rate
(intensity) , if
(1) X (t ) X (s ) is independent of F (s ) ;
(2) X (t ) X (s ) has the same probability distribution as
X (t s ) for 0 s t < 1 ; and
(3) X (t ) is
Announcements
Homework 2 (due Thursday).
Solution to Theoretical Parts is posted.
Please go to Discussion Sections and do quizzes.
Read Chapter 5 of the textbook.
1 / 18
Review
2 / 18
Poisson Process
A counting process X () is called a Poisson process wit
announcement
1 / 22
I
Homework 3 due Thursday night
I
Homework 3 theoretical solution is posted
I
Please go to the discussion section
I
Start reading Chapter 6.
Generalization of Poisson process
2 / 22
Compound Poisson Process
Suppose that fYk ; k 1g are
PSTAT 160 B Practice Final Winter 2017
180 minutes. Instructor: Tomoyuki Ichiba.
Write your name and all your answers clearly in the answer sheet.
1. Cars arrive to a single pump gas station according to a Poisson process with rate 3 cars per
hours. If a
PSTAT 160B Winter 2017
Homework 2
(1) The lifetime of A s dog and cat are independent exponential random variables with respective
rates rd and rc . One of them has just died. Find the expected additional lifetime of the other
pet.
Solution: Let us denote
PSTAT 160B Winter 2017
Homework 4
(1) A two dimensional Poisson process is a process of randomly occurring events in the plane such
that (i) for any region of area A the number of events in that region has a Poisson distribution
with mean A and (ii) the n
PSTAT 160B Winter 2017
Homework 1
(1) Suppose that you arrive at a single teller bank to find five other customers in the bank, one
being served and the other four waiting in line. You join the end of the line. If the service
times are all exponential wit
PSTAT 160B Winter 2017
Homework 7
(1) Suppose that the interarrival distribution for a renewal process is Poisson distributed with mean .
That is, suppose
k
; k = 0, 1, 2, . . .
P(Xn = k) = e
k!
(a) Find the distribution of Sn . (b) Calculate P(N (t) = n
PSTAT 160 B Practice Final Winter 2017
180 minutes. Instructor: Tomoyuki Ichiba.
Write your name and all your answers clearly in the answer sheet.
1. Cars arrive to a single pump gas station according to a Poisson process with rate 3 cars per
hours. If a
PSTAT 160B, Winter 2017. Homework 6
(1). A job shop consists of three machines and two repairmen. The amount of time a machine
works before breaking down is exponentially distributed with mean 10. If the amount of time it
takes a single repairman to fix a
PSTAT 160B Winter 2017
Homework 3
(1) Customers arrive at a bank at a Poisson rate . Suppose two customers arrived during the first two
hours. What is the probability that (a) both arrived during the first 20 minutes? (b) at least one arrived
during the f
PSTAT 160B Winter 2017
Homework 5
(1) What is the distribution of B(s) + B(t) for s t ?
Solution: Recall that B(t) can be written as
B(t) = (B(t) B(s) + B(s) ,
where B(s) N(0, s) , i.e., it is distributed in normal with mean 0 and variance s , independent
PSTAT 160 B Practice Midterm Winter 2017
75 minutes. Instructor: Tomoyuki Ichiba.
Write your name and all your answers clearly in the answer sheet.
1. [5 points each] Choose the correct answer.
1. Suppose that Z1 and Z2 are independent exponential random
Announcement
Overview: PSTAT 160B
This is a continuation from 160A. The topics covered in the
course are Continuous models; Continuous time stochastic
processes: Poisson process, Markov chains, Brownian motion,
simulation of these processes and their appl
PSTAT 160 B Practice Midterm Winter 2017
75 minutes. Instructor: Tomoyuki Ichiba.
Write your name and all your answers clearly in the answer sheet.
1. [5 points each] Choose the correct answer.
1. Suppose that Z1 and Z2 are independent exponential random
PSTAT 160B Winter 2017
Homework 3
Solve the exercises (1)-(10) below, and submit only the Python exercise (10) on Thursday February
2nd. The solution to (1)-(9) will be posted on Gauchospace.
(1) Customers arrive at a bank at a Poisson rate . Suppose two
1 / 20
Poisson Process
A counting process X () is called a Poisson process with rate
(intensity) , if
(1) X (t ) X (s ) is independent of F (s ) ;
(2) X (t ) X (s ) has the same probability distribution as
X (t s ) for 0 s t < 1 ; and
(3) X (t ) is distri
PSTAT 160B Winter 2016
Homework 5
Solve the exercises (1)-(9) below, and submit only the Python exercise (10) on Wednesday February
17th. The solution to (1)-(9) will be posted on Gauchospace.
(1) What is the distribution of 3(5) + 3(t) for s S t ?
Soluti
PSTAT 160 B Practice Final Winter 2016
180 minutes. Instructor: Tomoyuki Ichiba.
Write your name and all your answers clearly in the answer sheet.
1. Cars arrive to a single pump gas station according to a Poisson process with rate 3 cars per
hours. If a