section 1-a refresher on probability and statistics

8/8/2019 Section 1-A Refresher on Probability and Statistics

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A Refresher on Probability and

Statistics


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What Well Do ...

Review of probability and statistics necessary to do andunderstand simulation

Outline

Random variables

Hypothesis testing

Popular probability distributions


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Random Variables

One way of quantifying, simplifying events andprobabilities

A random variable (RV) is a number whose value is

determined by the outcome of an experiment Think: RV is a number whose value we dont know for sure

but well usually know something about what it can be or is

likely to be

Usually denoted as capital letters: X, Y, W1, W2, etc.

Probabilistic behavior described by distribution function


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Discrete vs. Continuous RVs

Two basic flavors of RVs, used to represent or modeldifferent things

Discretecan take on only certain separated values

Number of possible values could be finite or infinite

Continuouscan take on any real value in some range

Number of possible values is always infinite

Range could be bounded on both sides, just one side, or neither


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Discrete Distributions

LetXbe a discrete RV with possible values (range)x1,x2, (finite or infinite list)

Probability mass function (PMF)

p(xi) = P(X=xi) for i= 1, 2, ...

The statement X=xi is an event that may or may not happen,so it has a probability of happening, as measured by the PMF

Can express PMF as numerical list, table, graph, or formula

SinceXmust be equal to some xi, and since thexis are all

distinct,


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Cumulative distribution function (CDF)probability that theRV will be () a fixed valuex:

Properties of discrete CDFs

0 F(x) 1 for allx

Asx, F(x) 0

Asx +, F(x) 1

F(x) is non-decreasing inx

F(x) is a step function continuous from the right with jumps at

thexis of height equal to the PMF at thatxi

Discrete Distributions (contd.)

These four propertiesare also true ofcontinuous CDFs


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Discrete Distributions (contd.)

Computing probabilities about a discrete RVusually usethe PMF

Add up p(xi) for thosexis satisfying the condition for the event

With discrete RVs, must be careful about weak vs. strong

inequalitiesendpoints matter!


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Data set has a center the average (mean) RVs have a center expected value

Also called the mean or expectation of the RVX

Other common notation: m, mX Weighted average of the possible valuesxi, with weights being

their probability (relative likelihood) of occurring What expectation is not: The value ofXyou expect to get

E(X) might not even be among the possible valuesx1,x2,

What expectation is:Repeat the experiment many times, observe manyX1,X2, ,XnE(X) is what converges to (in a certain sense) as n

Discrete Expected Values


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Data set has measures of dispersion Sample variance

Sample standard deviation

RVs have corresponding measures

Other common notation:

Weighted average of squared deviations of the possible valuesxi from the mean

Standard deviation ofXis

Interpretation analogous to that for E(X)

Discrete Variances and

Standard Deviations


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Continuous Distributions

Now letXbe a continuous RV Possibly limited to a range bounded on left or right or both

No matter how small the range, the number of possible values

forXis always (uncountably) infinite

Not sensible to ask about P(X=x) even if x is in the possiblerange

Technically, P(X=x) is always 0

Instead, describe behavior of X in terms of its falling between

two values


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Continuous Distributions (contd.)

Probability density function (PDF) is a function f(x) with thefollowing three properties:

1. f(x) 0 for all real valuesx

2. The total area under f(x) is 1:

3. For any fixed a and b with a b, the probability thatXwill fallbetween a and b is the area under f(x) between a and b:

Fun facts about PDFs ObservedXs are denser in regions where f(x) is high

The height of a density, f(x), is not the probability of anythingit can even be > 1 With continuous RVs, you can be sloppy with weak vs. strong

inequalities and endpoints


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Continuous Distributions (contd.)

Cumulative distribution function (CDF) - probability that theRV will be a fixed valuex:

Properties of continuous CDFs

0 F(x) 1 for allx

Asx, F(x) 0

Asx +, F(x) 1

F(x) is non-decreasing inx

F(x) is a continuous function with slope equal to the PDF:

f(x) = F'(x)

These four propertiesare also true ofdiscrete CDFs

F(x) may ormay not havea closed-formformula


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Hypothesis Tests

Test some assertion about the population or itsparameters

Can never determine truth or falsity for sureonly getevidence that points one way or another

Null hypothesis (H0)what is to be tested

Alternate hypothesis (H1 or HA)denial ofH0H0:m= 6 vs. H1: m 6H0:s< 10 vs. H1:s 10H0:m1 = m2 vs. H1:m1 m2

Develop a decision rule to decide on H0

or H1

based onsample data


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Hypothesis Testing in Simulation

Input side Specify input distributions to drive the simulation

Collect real-world data on corresponding processes

Fit a probability distribution to the observed real-world data

Test H0: the data are well represented by the fitted distribution Output side

Have two or more competing designs modeled

Test H0: all designs perform the same on output, or test H0:

one design is better than another Selection of a best model scenario


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Popular Discrete Probability Distribution

Functions

Uniform Binomial

Geometric

Negative Binomial Poisson


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Discrete Uniform Distribution


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Binomial Distribution


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Negative Binomial Distribution


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Poisson Distribution


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Popular Continuous Probability Distribution

Functions

Uniform Normal

Triangular

Exponential Erlang

Weibull


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Continuous Uniform Distribution


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Normal Distribution


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Triangular Distribution


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Erlang Distribution


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Weibull Distribution

section 1-a refresher on probability and statistics

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