1STAT 13, UCLA, Ivo DinovSlide 1UCLA STAT 13Introduction toStatistical Methods for the Life and Health Sciences!Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology!Teaching Assistants:Ming Zheng, Annie CheUCLA StatisticsUniversity of California, Los Angeles, Winter 2004http://www.stat.ucla.edu/~dinov/courses_students.htmlSTAT 13, UCLA, Ivo DinovSlide 2Chapter 5: Discrete Random Variables!Random variables!Probability functions!The Binomial distribution!Expected valuesSTAT 13, UCLA, Ivo DinovSlide 3Definitions! An experiment is a naturally occurring phenomenon, a scientific study, a sampling trial or a test., in which an object (unit/subject) is selected at random (and/or treated at random) to observe/measure different outcome characteristics of the process the experiment studies.! A random variable is a type of measurement taken on the outcome of a random experiment.STAT 13, UCLA, Ivo DinovSlide 4Definitions! The probability function for a discrete random variable X gives P(X = x) [denoted pr(x) or P(x)]for every value x that the R.V. X can take! E.g., number of heads when a coin is tossed twicex 012pr(x )121414STAT 13, UCLA, Ivo DinovSlide 5Outcome GGG GGB GB BG BBG BBBProbability141818181814Stopping at one of each or 3 children! For R.V. X = number of girls, we haveX 0123pr(x )58181818Sample Space – complete/unique description of the possible outcomes from this experiment.STAT 13, UCLA, Ivo DinovSlide 6X 0123pr(x )58181818Plotting the probability function1230.75.50.25XP(X)2STAT 13, UCLA, Ivo DinovSlide 7! For each toss, P(Head) = p "P(Tail) = P(comp(H))=1-p! Outcomes: HH, HT, TH, TT! Probabilities: p.p, p(1-p), (1-p)p, (1-p)(1-p)! Count X, the number of heads in 2 tossesX 012pr(x )(1−p )22p (1−p )p2Tossing a biased coin twiceSTAT 13, UCLA, Ivo DinovSlide 8Hospital staysDays stayed x 45678910TotalFrequency 10 30 113 79 21 8 2 263Proportion pr(X = x) 0.038 0.114 0.430 0.300 0.080 0.030 0.008 1000CumulativeProportion0.038 0.152 0.582 0.882 0.962 0.992 1.000pr(X x)From Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.STAT 13, UCLA, Ivo DinovSlide 9123456789101112To get 4 to 8,and remove from 3 downpr(3 < X - 8)= pr(X - 8)pr(X - 3)[= pr(4 - X - 8)]x-values :start with everything up to 8Figure 5.2.2 Interval probabilities from cumulative probabilities. [This Figure represents an arbitrary distribution, not the hospital distribution.]aaFrom Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.Calculating Interval probabilitiesfrom cumulative probabilitiesP(3< X <9)P(X <9)P(X<=3)How to find the upper-tail?STAT 13, UCLA, Ivo DinovSlide 10Review! What is a random variable? What is a discrete random variable? (type of measurement taken on the outcome of random experiment)! What general principle is used for finding P(X=x)? (Adding the probabilities of all outcomes of the experiment where we have measured the RV, X=x)! What two general properties must be satisfied by the probabilities making up a probability function? (P(x)>=0; ) ! What are the two names given to probabilities of the form P(X ≤ x)?(cumulative & lower/left-tail)1)( =∑xxPSTAT 13, UCLA, Ivo DinovSlide 11Review! How do we find an upper/right-tail probability from a cumulative probability? [P(X>x) = 1-P(X<= x)]! When we use P(X ≤ 12) − P(X ≤ 5) to calculate the probability that X falls within an interval of values, what numbers are included in the interval? ([6:12])STAT 13, UCLA, Ivo DinovSlide 12Sample n balls and count X = # black balls in sampleM black ballsN – M white ballsN balls in an urn, of which there areThe two-color urn modelWe will compute the probability distribution of the R.V. X3STAT 13, UCLA, Ivo DinovSlide 13 Perform n tosses and count X = # headstoss 1 toss 2 toss npr(H) = ppr(H) = ppr(H) = pThe biased-coin tossing modelWe also want to compute the probabilitydistribution of this R.V. X!Are the two-color urn and the biased-coinmodels related? How do we present the models in mathematical terms?STAT 13, UCLA, Ivo DinovSlide 14! The distribution of the number of heads in ntosses of a biased coin is called the Binomial distribution.The answer is: Binomial distributionSTAT 13, UCLA, Ivo DinovSlide 15x0123456Individualpr(X = x)0.001 0.010 0.060 0.185 0.324 0.303 0.118Cumulative pr(X - x) 0.001 0.011 0.070 0.256 0.580 0.882 1.000Binomial(N, p) – the probability distributionof the number of Heads in an N-toss coin experiment, where the probability for Head occurring in each trial is p.E.g., Binomial(6, 0.7)For example P(X=0) = P(all 6 tosses are Tails) =001.03.0)7.01(66==−STAT 13, UCLA, Ivo DinovSlide 16Binary random processThe biased-coin tossing model is a physical model for situations which can be characterized as a series of trials where:#each trial has only two outcomes: success or failure;#p = P(success) is the same for every trial; and#trials are independent.! The distribution of X = number of successes (heads) in N such trials isBinomial(N, p)STAT 13, UCLA, Ivo DinovSlide 19Sampling from a finite population –Binomial ApproximationIf we take a sample of size n! from a much larger population (of size N)! in which a proportion p have a characteristic of interest, then the distribution of X, the number in the sample with that characteristic,! is approximately Binomial(n, p).$ (Operating Rule: Approximation is adequate if n / N< 0.1.)! Example, polling the US population to see what proportion is/has-been married.STAT 13, UCLA, Ivo DinovSlide 20Odds and ends …! For what types of situation is the urn-sampling model useful? For modeling binary random processes. When sampling with replacement, Binomial distribution is exact, where as, in sampling without replacement Binomial distribution is an approximation.! For what types of situation is the biased-coin sampling model useful? Defective parts. Approval poll of cloning for medicinal purposes. Number of Boys in 151 presidential children (90).! Give the three essential conditions for its applicability. (two outcomes; same p for every trial; independence)4STAT 13, UCLA, Ivo DinovSlide 21Odds and ends …! What is the distribution of the number of heads in ntosses of a biased coin?! Under what conditions does the Binomial distribution apply to samples taken without replacement from a finite population? When interested in assessing the distribution of a R.V., X, the number of observations in the sample (of n) with one
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