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# lec3print - Discrete random variables and probability...

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Discrete random variables and probability distributions Sayan Mukherjee Sta. 113 Chapter 3 of Devore August 30, 2007 Sayan Mukherjee Discrete random variables and probability distributions

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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 . Sayan Mukherjee Discrete random variables and probability distributions

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Intuition Think of a function as a machine. It has inputs and outputs: f : x y . A catapult Sayan Mukherjee Discrete random variables and probability distributions
Intuition The catapult takes as inputs: a rock and tension cord. The output is the distance the rock flies. Sayan Mukherjee Discrete random variables and probability distributions

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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 ? Sayan Mukherjee Discrete random variables and probability 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 ? Sayan Mukherjee Discrete random variables and probability distributions

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Discrete versus continuous rv Definition A discrete random variable is a rv which takes a finite or countable number of values. A continuous random variable is a rv which takes values in an interval of the real line or all of the real line. Sayan Mukherjee Discrete random variables and probability distributions
Bernoulli random variable Definition A random variable that takes values 0 or 1 is called a Bernoulli random variable . Sayan Mukherjee Discrete random variables and probability distributions

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Distribution function Definition The probability distribution function or probability mass function of a discrete random variable is defined for every possible x by p ( x ) = IP ( X = x ) = IP ( X ( s ) = x : for all s ∈ S ) . Sayan Mukherjee Discrete random variables and probability distributions
An example 0 5 10 15 20 0 0.05 0.1 0.15 0.2 x p(x) Sayan Mukherjee Discrete random variables and probability distributions

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Matlab code x= 1:20; y = poisspdf(x,4); plot(x,y,’*’) Sayan Mukherjee Discrete random variables and probability distributions
Some properties 1 p ( x ) 0 for all X = x 2 x p ( x ) = 1 Sayan Mukherjee Discrete random variables and probability distributions

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Parameter of a pdf Definition If p ( x ) is parameterized by a quantity α then α is the parameter of the pdf and the set of pdfs characterized by varying α is called a family of distribution functions.
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lec3print - Discrete random variables and probability...

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