CS457 - System Performance Evaluation - Winter 2008


Questions and Comments

  1. Tutorial: MC4045, Monday March 10, 18.00 to 19.00
  2. Ass3. Exponential distribution with mean interarrival time 1/k means distributed by kexp(-kt)

    Similar for mean service time.


Lecture 23 - Useful Distributions for Random Numbers

Text: chapter 28 & 29

Random Variables

5. Binomial Distribution

Get exactly x correct in n tries.

There is no way of doing the sum as we did for the geometric distribution,

So, generate n Bernoulli variables and sum them.

6. Zipf Distribution

Order a set of N numbers.

The probability that the k'th number occurs is k^-s.

In the limit N-> infinity this is the Rieman zeta-function

Continuous Random Variables

1. Arbitrary distribution

Use the inverse transformation method.

The easiest way is to interpolate linearly in a table to solve F(x) = u

2. Exponential distribution

Use for infinite populations that produce finite event rates.

  1. f(x) = \lambda exp(-\lambda*x)
  2. F(x) = 1 - exp(-\lambda*x)
  3. u = F(x)

    exp(-\lambda*x) = 1 - u

    x = -log(1-u) / \lambda = -log(u) / \lambda

    But, remember that there is a special case! (u = 0)

3. Continuous uniform

Use when you don't know anything, except possibly theoretically.

Interpolation on the importance of prior distributions

  1. f(x) = 1/(b-a) for a<x<b
  2. F(x) = (x-a) / (b-a)
  3. x = a + u*(b-a)

4. Pareto distribution

Use for heavy-tailed data (also called self-similar)

  1. f(x) = ax^-(a+1)
  2. F(x) = 1 - x^-a
  3. x = (1-u)^-(1/a)

5. Normal distribution

Use when you want to add noise

  1. f(x) = exp(-(x^2)/2*\sigma^2)
  2. No easy form for F(x).
  3. But, f(x,y) = exp(-(x^2 + y^2) / 2*\sigma) has a simple F(x,y), which I won't work out.
  4. Thus, you choose two uniform variates, u1 & u2.
  5. And you get back two independent random numbers

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