04 Discrete Probability - Discrete Probability...

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Discrete Probability Distributions Devore and Berk Chapter 3
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What is a random variable? { Used to help characterize the probability of events { Defined as a function that maps sample space to real numbers z And we can do mathematics on real numbers! { Can be discrete or continuous { Often represented by uppercase letters, so X random, x realized
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Example { X = Gender of a randomly selected student from the class z Can be either Male or Female, so the sample space is the set {Male, Female} z An event: Female { So, we select a student, and it is a Male, then x=Male
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Types of Random Variables { Discrete z Result from a single coin toss z Number of heads from 12 coin tosses z Sum of points from a throw of two dice z Number of individuals in a fish population { Continuous z Length of a lifetime z Distance between trees
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Probability Distribution { A probability distribution describes how probability is distributed among possible values { Fair coin: P(X=H)=1/2,P(X=T)=1/2 z Flip coin, realization: x=H { Die, one toss, P(X=4)=1/6 z Toss die, realization: x=3
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123456 0.0 0.05 0.10 0.15 Theoretical Distribution 0 50 100 150 x.bar Realized Empircal Distribution
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Script to explore probability of a single role of a die par(mfrow=c(2,1)) barplot(c(1/6,1/6,1/6,1/6,1/6,1/6),names=c(1,2,3,4,5,6),cex=1.2) title("Theoretical Distribution") x = runif(1000) x.bars = cut(x,breaks=c(0,1/6,2/6,3/6,4/6,5/6,6/6)) hist(as.character(x.bars),names=c(1,2,3,4,5,6),xlab="x.bar",cex=1.2) title("Realized Empircal Distribution") par(mfrow=c(1,1))
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Cumulative Probability Distribution = = x y y y p x X P x F : ) ( ) ( ) (
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Number of Points on a Die Cumulative Probability 0246 0.0 0.2 0.4 0.6 0.8 1.0 Cumulative Probability, P(X<x)
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Expected Value = = D x x p x X E ) ( ) ( μ
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Expected Value from a Single Throw of a Die 5 . 3 6 6 1 5 6 1 4 6 1 3 6 1 2 6 1 1 6 1 = + + + + + = μ
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Expected Value of a Function = D x x p x h X h E ) ( ) ( )) ( (
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04 Discrete Probability - Discrete Probability...

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