Lecture06 - 01 May 2003 Biostatistics 6650--L6 1 Todays...

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Unformatted text preview: 01 May 2003 Biostatistics 6650--L6 1 Todays Schedule Continuous random variables Probability density function Cumulative distribution function Normal Distribution Standard Normal Distribution Calculating probabilities Conversion Normal approximation to Binomial/Poisson Handout Normal Tables 01 May 2003 Biostatistics 6650--L6 2 Continuous Random Variables A RV whose possible values cannot be enumerated is a continuous RV. X= height(cm) for 5th graders all unique if measured precisely X= PSA values at time of diagnosis of prostate cancer X= cholesterol level X= time from diagnosis to recurrence 01 May 2003 Biostatistics 6650--L6 3 a b f(x) X D e n s i t y , f ( x ) Continuous Random Variables Probability density function(pdf) The pdf of the random variable X is a function, f(x), such that the total area under the curve is 1, and the area between any two points, X=a,X=b, is equal to the probability that X is between a and b. PDF takes on high values in regions of high probability Pr(a<X<b) 01 May 2003 Biostatistics 6650--L6 4 Continuous Random Variables Cumulative distribution function(cdf) The cdf for a continuous random variable X evaluated at point a, denoted as F(a), is defined as the Pr(X< a). Pr(X=a)=0 Pr(X< a) = Pr(X<a) + Pr(X=a) = Pr(X<a) Pr(a<X<b)=Pr(a< X<b) =Pr(a<X< b) =Pr(a< X< b) 01 May 2003 Biostatistics 6650--L6 5 1 Continuous Random Variables Cumulative distribution function(cdf) --new F(a), is defined as the Pr(X< a). Pr(a<X<b) = Pr(X<b) - Pr(X<a) = F(b)-F(a) F(x) is used extensively in calculation of p-values 1-F(c)=Pr(X>c) = probability of a result as extreme as c or greater a b F ( X ) c F(c) F(a) X F(b) a b f(x) X D e n s i t y , f ( x ) c 01 May 2003 Biostatistics 6650--L6 6 Continuous Random Variables Expected value of continuous RV X, denoted by E(X) or is average value in the population The variance of a continuous RV X, denoted by Var(X) or Standard deviation= Same meaning as for discrete X Mathematical derivation for continuous X requires calculus X 2 X 2 2 ( ) X X E X =- ( ) X Var X = 01 May 2003...
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Lecture06 - 01 May 2003 Biostatistics 6650--L6 1 Todays...

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