Summer 2007 STAT333 Summary of the waiting time random variable r.v.
Let E be an event and TE be the waiting time for the 1st E. Note that in general, the possible range of TE is R = cfw_1, 2, 3, . . . cfw_, where cfw_ means we can not observe E. We are i
STAT 333 Tutorial 8
Nov. 29, 2010
1. Customers enter a department store according to a Poisson process cfw_N (t), t 0 at rate = 10 per hour. Of the entering customers, suppose that 70% of them are female and the remaining 30% are male. Assume that custome
Syllabus for STAT 333: Applied Probability Fall 2010 Instructor: Zhongxian (Chris) Men Office: MC 6081A E-mail: zmen@math.uwaterloo.ca Note: Please contact me through Angel e-mail or if you want send email to my uwaterloo
account. I will only answer email
Summer 2007 STAT333 Summary of the exponential distribution and the Poisson process
Exponential distribution: (a) pdf function: f (x) = e-x , x > 0 and > 0; (b) tail probability: P (X > x) = e-x ; (c) expectation=1/ and variance=1/2 ; (d) no-memory prope
Summer 2007 STAT333 Summary of the continuous Markov chain
We already had a summary of the discrete Markov Chain. Almost all of definitions and results in the discrete Markov chain can be extended to the continuous Markov chain. Let me just review some sp
STAT 333 Test 1 SOLUTION
October 5, 2010
1. [5] In a small factory, there are 4 machines which package the same product. The percentages that these machines package are 15%, 20%, 30% and 35%. Also the probabilities that these machines can damage the produ
Results for Some Fundamental Probability Distributions
Discrete Distribution Probability Mass Function of X Mean E(X) Variance Var(X)
Binomial(n, p)
p(x) =
n x
px (1 - p)n-x , x = 0, 1, ., n
np
np(1 - p)
Bernoulli(p)
p(x) = px (1 - p)1-x , x = 0, 1
r (x)(
Stat 333 Summer 2007 Renewal theory: Practice Problems
1. Below are some hypothetical first waiting-time distributions fn = P (T = n) where is a renewal event of a stochastic process. In each case determine whether is transient or recurrent; if recurrent,
Stat 333 Summer 2007 Renewal theory: Practice Problems
1. Below are some hypothetical first waiting-time distributions fn = P (T = n) where is a renewal event of a stochastic process. In each case determine whether is transient or recurrent; if recurrent,
Summer 2007 STAT333 Summary of the Random walk
Random walk is a simple stochastic process, which can be regarded as an example for the renewal process, and can also be regarded as an example for the discrete Markov process. The following are some formulas
Summer 2007
STAT333
Summary of generating function
The purpose of this part is two-folds. First, given a sequence of real numbers cfw_an , how n=0 to calculate the generating function A(s). Secondly, given the generating function A(s), how to find the co
Stat 333 Summer 2007 Probability generating function, Branching process Practice Problems
READING: Ross's textbook, Page 233 to Page 236. The following are some problems regarding the generating function, the probability generating function and the branch
Stat 333 Summer 2007 Markov Chain: Practice Problems
The following problems are from Chapter 4 of Ross (plus a few extra thrown in). The problem numbers are the same in both the 8th and 9th editions. Solutions will be provided for most of these problems.
Summer 2007 STAT333 Summary of the Markov Chain
Suppose we have a sequence of random variables, cfw_X1 , X2 , . . . = cfw_Xn , which is also n=1 called a stochastic process. Markov Chain. (a) State space (S): all the possible values of cfw_Xn . n=1 (b)
STAT 333 Fall 2010 October 18, 2010 [Thursday, Sep 30, 2010] Objective: Conditional expectation and double-average theorem Example 1 Consider i.i.d. Bernoulli trials with pr (S) = p, 0 < p < 1. Define that Ts = the waiting time for the first S, and Tss =
STAT333 Summary of the branching process
The branching process is the first example of simple stochastic processes. Suppose we have cells that are capable of producing cells of like kind. u Z0 = 1
d d u du u u d d u u u u du rr d drr u r u u u u du r u
STAT 333 Assignment 3
Due: Monday, Dec. 6 in MC6095 between 1:00 to 3:00
(Please print) Last name: First Name: Acknowledgements: ID#:
Mark:
TA's initials:
1. Consider a random walk on the integers, starting at 5. The random walk jumps to the right with pr
STAT 333 Assignment 2
Due: Thursday, Nov. 4 at the beginning of the class
(Please print) Last name: First Name: Acknowledgements: ID#:
Mark:
TA's initials:
1. let Xi , i = 1, 2, .n, are i.i.d. random variables with uniform distributions on [0, 1], where n
STAT 333 Assignment 1
Due: Thursday, Sep. 30 at the beginning of the class
(Please print) Last name: First Time: Acknowledgements: ID#:
Mark:
TA's initials:
Chapter 1 1. (#18)Assume that each child who is born is equally likely to be a boy or a girl. If a