About ten minutes late starting up again. Test next week. Covering counting methods today. Got a copy of the first test over this material from 2004; that should be helpful. Test is open book, open note. That may help or mey be a trap.
Urn problems — Imagine an urn filled with items. Urn problems are concerned with how many ways one could randomly select items from that urn.
Important things to consider:
- Is the order in which items are selected considered or not considered?
- Are items being replaced between trials?
The number of elements in the sample space matters greatly. The combination of the total number of elements and the manner in which order and replacement are handled determines the number of possible results.
In general, combinations of results do not concern order. Permutations do concern order.
Permutations, in which order matters and items are not replaced, number N!/(n-x)!, where x is the number of items chosen from, and n is the number of items chosen.
Chapter 5: Concerned with probability distributions of discrete random variables
Chapter 6: Concerned with probability distributions of continuous random variables
Discrete variables have only specific values as valid entries. The set of counting numbers is the most common set of discrete numbers.
A continuous variables can have any value within its possible range, and therefore always has infinite potential values. Discrete variable may or may not have infinite possible values.
Bernoulli (Binary Distribution) — Has two possible values. 0 and 1. The simplest discrete variable setup that conveys any information.
Binomial — The sum of independent, identical Bernoulli variables. Success or failure.
Poisson
Hypergeometric
Of the above four items, we must be able to calculate probabilities, expected values, population mean, variance, and standard variation.
If you have a function y = f(x), the expected value of y is the sum of all the occurences of x times the probability of those specific occurences of x.
The expected value of a function is also represented by the greek letter mu. The expected value is the long-term average outcome.