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Lecture05 - Todays Schedule Discrete random variables...

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25 April 2003 Biostatistics 6650--L5 1 Today’s Schedule Discrete random variables Definition/examples Probability distribution function Cumulative distribution function Population mean/variance Permutations/Combinations Binomial distribution Poisson distribution Handout Binomial/Poisson Tables.

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25 April 2003 Biostatistics 6650--L5 2 Discrete Random Variables Definition A random variable(RV) for which there exist a discrete set of values with specified probabilities is said to be a discrete RV . X= number of days of the week with a sentinel event(0,1,…,7) X= outcome of a coin flip, H or T X= sex of a newborn child, M or F X= number of episodes of otitis media in first 24 months (0,1,2,….)
25 April 2003 Biostatistics 6650--L5 3 Discrete 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

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25 April 2003 Biostatistics 6650--L5 4 Discrete Random Variables Probability distribution function(pdf) Rule that assigns to any possible value of a discrete RV X, the probability that X=x or Pr(X=x), where x is the observed value. ( Sometimes r is used instead of x.) Examples X= number of heads in 3 coin flips X=x 0 1 2 3 Pr(X=x) 1/8 3/8 3/8 1/8 X= gender of newborns X=r Female Male Pr(X=r) .485 .515
25 April 2003 Biostatistics 6650--L5 5 Discrete Random Variables Cumulative distribution function(cdf) The cdf of a random variable X, is denoted by F(X), and for a specific values of X=x, by F(x), where F(x)=Pr(X<=x). Examples X= number of heads in 3 coin flips X=x 0 1 2 3 Pr(X=x) 1/8 3/8 3/8 1/8 F(x) 1/8 4/8 7/8 8/8

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25 April 2003 Biostatistics 6650--L5 6 Discrete Random Variables Cumulative distribution function(cdf) Example X= episodes of otitis media in first 24 mos x Pr(X=x)* F(x)=Pr(X<=x) 0 .129 .129 1 .264 .393 2 .271 .664 3 .185 .849 4 .095 .944 5 .039 .983 6+ .017 1.000 *Hypothetical.
25 April 2003 Biostatistics 6650--L5 7 Discrete Random Variables Cumulative distribution function(cdf) step function ranges from 0 to 1 most intuitive with ordinal or count discrete RV’s Theoretical versus sample pdf and cdf If we know the population(theoretical) pdf/cdf under a given model(fair coin, say), we can assess the model fit by comparing the sample pdf and cdf to the theoretical values If the model is correct , as we take larger and larger samples, the sample pdf/cdf will approach the population values Role of statistical inference is to determine if differences between the sample and the model are real, or within the realm of chance. Rosner

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25 April 2003 Biostatistics 6650--L5 8 Discrete Random Variables Theoretical versus sample pdf Example: Seasonality of hip fractures X=the season of hip fractures Sample for 1 year, observe 50 fractures Tally results by season Assume null model--no effect of season Season(r) Pr(X=r) Sample Freq. Distn Dec/Jan/Feb .25 .5=25/50
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