Sampling and Descriptive
Statistics
1
What is statistics?
Statistics is the science of data.
This involves collecting, classifying, summarizing,
organizing, analyzing and interpreting numerical
information.
Statistics
Descriptive
Statistics
2
Inferential
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 level, and
household incomes or student grades, etc
Nor
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 be processed.
Examples
A personal or shared computer ex
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-state
Two classes of counting processes
2
counts of arr
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
variable is countable
Let X be a random variable representi
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 of
snowy days (in the next 3 days)?
Given that today i
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
Absorbing
3
Transient
Null recurrent
Non-absorbing
Recurrent v
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 network
Number of customers in a supermarket
The quantity
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 first
success has Geometric distribution.
2
Examples
A sea
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( X , Y ) E( X E( X )(Y E(Y )
Alternate formula:
Cov(X,
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
Outcome of an
experiment
X f ( )
Real Number
The distri
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 different order produce the
same outcome.
nCk
or
n
n!
k k!(
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 card
2
Sample point
A sample point is the most basic outco
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 nine was
surprised by everything. 'If you don't
shut you