THE DESTRUCTORS
Graham Greene
It was on the eve of August Bank
Holiday that the latest recruit became
the leader of the Wormsley Common
Gang. No one was surprised except
Mike, but Mike at the age of n
Normal distribution
1
Normal distribution
The most commonly used distribution in statistics.
Often found to be a good model for physical variables like
weight, height, temperature, voltage, pollution
Queuing Systems
1
Queuing system
A queuing system is a server facility consisting of
2
One or several servers designed to perform certain tasks or
process certain jobs, and
A queue of jobs waiting to
Poisson Process
1
Counting Process
A stochastic process () is counting if () is the
number of items counted by time t.
() is always non-decreasing with = cfw_0,1,2,3,
Counting processes are discrete-
Continuous distribution
1
Continuous distribution
A continuous random variable is one which can take a
continuous range of values
discrete distribution - the set of possible values for the random
vari
Markov Chain (Part 3)
1
Outline
Expected number of return
Recurrence and transience
Absorbing Markov Chains
2
Example: Weather example (revisited)
Given that today is nice, what is the expected number
Markov Chain (Part 2)
1
Outline
Classification of states
Existence of equilibrium distribution and limit distribution
Random walk
2
Classification of States
States
Recurrent
Positive recurrent
Absorbi
Markov Chain (Part 1)
1
Motivation
We are interested in randomness which evolve with time
Examples
2
Length of queues
Number of students passing a course
Temperature outside
Number of packets in a net
Discrete Distributions
1
Geometric Distribution
Consider a sequence of independent Bernoulli trials, each
with the same probability of success, p.
The number of Bernoulli trials needed to get the fir
Random variables
1
Covariance
Expectation, variance, and standard deviation characterize
the distribution of a single random variable.
Measures the linear relationship between two random
variables
Cov
Random variables
1
Random variable
A random variable is a variable that assumes numerical
values associated with the random outcome of an
experiment
Maps an outcome of an experiment to a real number
O
Probability
1
Combination
Unordered subset of size k from n objects
The number of ways of choosing k object from a group of
n objects
Difference from permutation:
The same objects sampled in a differe
Probability
1
Experiment
An experiment is an act or process of observation that
leads to a single outcome that cannot be predicted with
certainty
Examples:
rolling a die
tossing a coin
drawing a car
Sampling and Descriptive
Statistics
1
What is statistics?
Statistics is the science of data.
This involves collecting, classifying, summarizing,
organizing, analyzing and interpreting numerical
inform