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Unformatted text preview: Continuous random variables Continuous distributions Continuous random variables and probability distributions Sayan Mukherjee Sta. 113 Chapter 4 of Devore September 13, 2007 Sayan Mukherjee 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 Chisquared Beta Sayan Mukherjee 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 vales is an entire interval of real numbers: x [ A , B ] for A < B. Sayan Mukherjee Continuous random variables and probability distributions Continuous random variables Continuous distributions Examples 1 Heights of people. Why is it that the children of very tall people tend to be shorter than their parents ? 2 Amount of rainfall per square meter. 3 Snowflakes (complex continuous random variables). Sayan Mukherjee 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 ) = Z 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. Sayan Mukherjee Continuous random variables and probability distributions Continuous random variables Continuous distributions Picture Sayan Mukherjee 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 ) . Sayan Mukherjee 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 ) = Z x p ( u ) du . So for for each x, F ( x ) is the area under the density to the left of x. Sayan Mukherjee Continuous random variables and probability distributions Continuous random variables Continuous distributions Picture108642 2 4 6 8 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sayan Mukherjee 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); Sayan Mukherjee Continuous random variables and probability distributions Continuous random variables Continuous distributions More properties Proposition Let X be a continuous rv with pdf p...
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 Fall '08
 MUKHERJEE
 Statistics, Probability

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