Permutations and Combinations
Combination Formula
Conditional Probability and Bayes' Rule
Odds
Discrete Random Variables
Permutations
X!
Classical Balls Problem
Add the balls and divisions together as objects:
Probability Language
Sample Space
The set of all possible outcomes in a situation.
An element is a sample point. A subset of a sample space is an event.
Mutually Exclusive
Events are said to be mutually exclusive if they are independent of each other.
Probability Laws
De Morgan's Laws
Distributive Law
Binomial Distribution
Poisson Distribution
Geometric Distribution
Negative Binomial Distribution
Decision Problems
- Alternative decisions: decision variables D
- The environment: environmental variables X
- Outcomes: indicated by D, X pairs
- Utilities: R(D, X)
- Objective (max, min, optimise): max(d)R(D, X)
- Constraints limits: e.g. D < 10
Decision Trees
Payoff Tables
- Rows are decisions (D)
- Cols are environment variables (X)
Continus Random Variables
- The set of values it can take is not countable
- It has a PDF in which its CDF can be obtained from
EMV
EVPI = EMV(perfect) - EMV
EVSI = EMV(survey) - EMV
ENGSI = EVSI - cost of survey
Uniform Distribution
Normal Distribution
Exponential Distribution
Beta Distribution
Cumulative Density Function
- The function is a non-decreasing function
- As x approaches negative infinity Fx(x) approaches or equals 0
- As x approaches positive infinity Fx(x) approaches or equals 1
Probability Density Function
- f(x) is non-negative for all of x
- Its area under the curve from negative infinity to positive infinity is equal to one
Estimation and Goodness of Fit
Estimation and Error Bounds
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