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# lec4print - Continuous random variables Continuous...

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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

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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

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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

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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

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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 0 2 4 6 8 10 0 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

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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 < b, IP ( a X b ) = F ( b ) - F ( a ) .

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