# CS457 - System Performance Evaluation - Winter 2008

1. Midterm: Q5e
2. Assignment 3

# Lecture 22 - Random Numbers

Text: chapter 12 28 & 29

## Random Variables

The mathematical foundation for random number is the random variable, which is a variable that has different values on different occasions.

1. A probability distribution determines what values a random variable gets, and how often
2. Discrete versus continuous random variables
3. We usually assume that the random variables we want are independent and identically distributed (iid or IID).
• Why is this assumption reasonable?
• Think about where we get the distributions from.
4. Most random variables we use have underlying distributions that are non-negative.
• Why?
5. Random variables are normally written as capitals.

#### Random variable concepts

1. Cumulative distribution function (cdf or CDF):
• F(x) = P(X <= x)
• F(0) = 0
• F(infinity) = 1
2. Probability density function (pdf or PDF):
• Continuous distributions only
• F(x) = \int_0^x f(x') dx'
• Mean: \mu = E(X) = \int_0^\infinity x'f(x')dx'
• Variance: \sigma^2 = X((X-E(X))^2) = \int_0^\infinity (x'-E(X))^2 f(x')dx'
3. Probability mass function (pmf or PMF):
• Discrete distributions only: x takes a value fro a countable set {x1, x2, x3, ...} where ... may or may not terminate
• P(X = xi) = pi
• \sum_i pi = 1
• F(x) = \sum_(xi<=x) pi
• Mean: E(X) = \sum_i xi pi
• Variance: \sum_i (xi - E(X) )^2 pi

In discrete event simulation we model quantities like interarrival time or service time as IID random variables.

As a result we need to generate random variables from random number generators.

• Random number generators normally produce random floating point numbers uniformalu distributed between 0 and 1. (Technically, in the range [0.0..1.0).)
• We need to know how to transform this output into the distribution we want.

#### Random variables with arbitrary distributions

A random variable is uniquely defined by its CDF, F(x).

• 0 <= F(x) <= 1
• Suppose we want to express this random variable as a transformation Y = g(X) of a uniformly distributed random variable.
• P(Y < y) = Fy(y) = P(X < g^-1(y) ) = Fx( g^-1(y) ).
• Choose g(x) = Fx(x). Then P(Y < y) = y, or fy(y) = 1. Y is uniformly distributed.

#### Examples of Discrete Distributions

1. Bernoulli
2. Discrete uniform
3. Geometric Discrete
4. Binomial
5. Poisson