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
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