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
A queuing system is a server facility consisting of
One or several servers designed to perform certain tasks or
process certain jobs, and
A queue of jobs waiting to be processed.
A personal or shared computer ex
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
counts of arr
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)
Expected number of return
Recurrence and transience
Absorbing Markov Chains
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)
Classification of states
Existence of equilibrium distribution and limit distribution
Classification of States
Markov Chain (Part 1)
We are interested in randomness which evolve with time
Length of queues
Number of students passing a course
Number of packets in a network
Number of customers in a supermarket
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.
Expectation, variance, and standard deviation characterize
the distribution of a single random variable.
Measures the linear relationship between two random
Cov( X , Y ) E( X E( X )(Y E(Y )
A random variable is a variable that assumes numerical
values associated with the random outcome of an
Maps an outcome of an experiment to a real number
Outcome of an
X f ( )
Unordered subset of size k from n objects
The number of ways of choosing k object from a group of
Difference from permutation:
The same objects sampled in a different order produce the
An experiment is an act or process of observation that
leads to a single outcome that cannot be predicted with
rolling a die
tossing a coin
drawing a card
A sample point is the most basic outco
Sampling and Descriptive
What is statistics?
Statistics is the science of data.
This involves collecting, classifying, summarizing,
organizing, analyzing and interpreting numerical