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Unformatted text preview: 01 May 2003 Biostatistics 6650L6 1 Today’s 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 6650L6 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 6650L6 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 6650L6 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 6650L6 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 pvalues • 1F(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 6650L6 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 μ 2 X σ 2 2 ( ) X X E X σ μ = ( ) X Var X σ = 01 May 2003...
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This note was uploaded on 06/17/2011 for the course BME 6650 taught by Professor Multipleinstructors during the Spring '03 term at Mayo Clinic College of Medicine.
 Spring '03
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