Transcript Review of Basic Probability and Statistics

Review of Basic Probability and
Statistics
ISE525: Spring 10
Random Variables and Their Properties
• Experiment : a process whose outcome is not
known with certainty.
• Set of all possible outcomes of an experiment is
the sample space.
• Outcomes are sample points in the sample space.
• The distribution function (or the cumulative
distribution function, F(x), of the random variable
X is defined for each real number as follows:
Properties of distribution functions
1)
2) F(x) is nondecreasing.
3)
Discrete Random Variables
• A random variable, X, is said to be discrete if it
can take on at most a countable number of
values:
• The probability that X takes on the value xi is
given by
• Also:
• p(x) is the probability mass function.
Discrete variables continued:
• The distribution function F(x) for the discrete
random variable X is given by:
Moments
Moments of a Probability Distribution
• The variance is defined as the average value of
the quantity : (distance from mean)2
• The standard deviation, σ =
For discrete Random Variables
Continuous random variables
• A random variable X is said to be continuous if
there exists a non-negative function, f(x), such
that for any set of real numbers B,
• Unlike a mass function, for the continuous
random variable, f(x) is not the probability
that the random number equals x.
Multiple random variables
• IF X and Y are discrete random variables, then
the joint probability mass function is:
• P(x,y) = P(X=x, Y=y)
• X and Y are independent if:
Multiple random variables
• For continuous random variables, the joint pdf
is
• For independence:
Properties of means
• This holds even if the variables are
dependent!
Properties of variance
• This does not hold if the variables are
correlated.
Common Discrete Distributions
• Bernoulli: Coin toss
• Binomial: Sum of Bernoulli trials
• Poisson Distribution:
Common Continuous Distributions
• Uniform
Exponential Distribution
• Probability distribution function (pdf) and the
Cumulative distribution functions (cdf) are:
• Mean and Standard Deviation are:
Common Continuous Distributions
• Normal Distribution:
Other Distributions
• Erlang distribution:
Gamma Distribution
Estimation
• Means, variances and correlations:
• Simulation data are almost always correlated
(according to Law and Kelton) !
Hypothesis tests for means
Strong law of large numbers