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