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lec4print - Continuous random variables Continuous...

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Unformatted text preview: Continuous random variables Continuous distributions Continuous random variables and probability distributions Artin Armagan Sta. 113 Chapter 4 of Devore January 28, 2009 Artin Armagan Continuous random variables and probability distributions Continuous random variables Continuous distributions Table of contents 1 Continuous random variables 2 Continuous distributions Uniform Normal Exponential Gamma Chi-squared Beta Artin Armagan Continuous random variables and probability distributions Continuous random variables Continuous distributions Mathematical definition Definition A random variable X is continuous if its set of possible values is an entire interval of real numbers: x [ A , B ] for A < B. Artin Armagan Continuous random variables and probability distributions Continuous random variables Continuous distributions Examples 1 Heights of people. 2 Amount of rainfall per square meter. 3 IQ scores. Artin Armagan Continuous random variables and probability distributions Continuous random variables Continuous distributions Probability distributions Definition Let X be a continuous rv. The probability density function (pdf) of X is a function p ( x ) such that for any two numbers a b IP ( a X b ) = integraldisplay b a p ( x ) dx . The probability that X takes values in the interval [ a , b ] is the area under the graph of the density function in the interval. Artin Armagan Continuous random variables and probability distributions Continuous random variables Continuous distributions Picture Artin Armagan Continuous random variables and probability distributions Continuous random variables Continuous distributions Restatement Proposition Let X be a continuous rv. Then for any number c, IP ( X = c ) = 0 and for any two numbers a < b IP ( a X b ) = IP ( a < X b ) = IP ( a X < b ) = IP ( a < X < b ) . Artin Armagan Continuous random variables and probability distributions Continuous random variables Continuous distributions Cumulative distribution function Definition The cumulative distribution function (cdf) F ( x ) for a continuous rv X is defined for every number x by F ( x ) = IP ( X x ) = integraldisplay x p ( u ) du . So for for each x, F ( x ) is the area under the density to the left of x. Artin Armagan Continuous random variables and probability distributions Continuous random variables Continuous distributions Picture-10-8-6-4-2 2 4 6 8 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Artin Armagan Continuous random variables and probability distributions Continuous random variables Continuous distributions Matlab code x= -10:.01:10; y = normpdf(x,.5,1); plot(x,y,r); hold on; y1 = normcdf(x,.5,1); plot(x,y1,b); Artin Armagan Continuous random variables and probability distributions Continuous random variables Continuous distributions More properties Proposition Let X be a continuous rv with pdf p ( x ) and cdf F ( x ) . Then for any number a, IP ( X > a ) = 1- F ( a ) , and for any two numbers a <...
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This note was uploaded on 01/17/2010 for the course STA 113 taught by Professor Staff during the Fall '08 term at Duke.

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lec4print - Continuous random variables Continuous...

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