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Random variables Distributions for discrete random variables Expectation and variance Discrete distributions Discrete random variables and probability distributions Artin Armagan Sta. 113 Chapter 3 of Devore January 16, 2009 Artin Armagan Discrete random variables and probability distributions
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Random variables Distributions for discrete random variables Expectation and variance Discrete distributions Table of contents 1 Random variables 2 Distributions for discrete random variables 3 Expectation and variance 4 Discrete distributions Bernoulli Binomial Hypergeometric Poisson Artin Armagan Discrete random variables and probability distributions
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Random variables Distributions for discrete random variables Expectation and variance Discrete distributions Mathematical definition Definition A random variable is a function that maps an event from the sample space S to a real number: X : ω R , where ω ∈ S . Artin Armagan Discrete random variables and probability distributions
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Random variables Distributions for discrete random variables Expectation and variance Discrete distributions Intuition Think of a function as a machine. It has inputs and outputs: f : x y . A catapult Artin Armagan Discrete random variables and probability distributions
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Random variables Distributions for discrete random variables Expectation and variance Discrete distributions Intuition The catapult takes as inputs: a rock and tension cord. The output is the distance the rock flies. Artin Armagan Discrete random variables and probability distributions
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Random variables Distributions for discrete random variables Expectation and variance Discrete distributions Intuition The machine/function is now the flipping of a quarter by my thumb. The output is one of two possibilities: { H , T } . Let us call H = 1 and T = 0. This function is a (discrete) random variable it maps { H , T } into real numbers { 0 , 1 } , Why is this function random ? Why is it discrete ? Artin Armagan Discrete random variables and probability distributions
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Random variables Distributions for discrete random variables Expectation and variance Discrete distributions Intuition Back to the catapult. Even if we know exactly the size and shape of the rock as well as the tension of the cord, the distance the rock flies may not always be the same due to variation in wind and many other factors. The catapult is a (continuous) random variable it maps the state of the catapult to real numbers [0 , ). Why is this function random ? Why is it continuous ? Artin Armagan Discrete random variables and probability distributions
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Random variables Distributions for discrete random variables Expectation and variance Discrete distributions Discrete versus continuous rv Definition A discrete random variable is a rv which takes a finite or countable number of values.
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