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# Statistical Analysis STAT 401

UI

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This 12 page Class Notes was uploaded by Mr. Alex Berge on Friday October 23, 2015. The Class Notes belongs to STAT 401 at University of Idaho taught by Brian Dennis in Fall. Since its upload, it has received 14 views. For similar materials see /class/227936/stat-401-university-of-idaho in Statistics at University of Idaho.

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Date Created: 10/23/15

CHAPTER 4 PROBABILITY AND PROBABILITY DISTRIBUTIONS A B any two events A is the complement of A Additive law PAorB PA PB PAandB Multiplicative law PAandB PABPB PBAPA Independence A and B are independent if P AIB P A Bayes39 formula PAB P 3133 A This is Bayes39 formula Often the denominator probability must be obtained as a sum of joint probabilities for instance the event B can be written as the union of B and A and B and A So PB PB andA PB and Z PBAPA PBZPZ and PWB Z ltBAgtPltAgt P PBAPA PBZPZ 39 ex A drawer has 2 white socks and 2 blue socks Professor reaches in and draw out 2 socks in succession without replacement W1 2 white on the rst draw W2 2 white on the second draw 31 2 blue etc A tree diagram of the events and the conditional probabilities is W2 PW2W1 W1 PB2W1 B2 PW1 Z P31 W2 PltW2IBlgt Bl PB2Bl B2 Probability of two white socks PW1andW2 PW2W1PW1 g 2 Probability of two socks the same color LAM a Ol Ol PW1 and W2 P31 and 32 2 Probability that the second sock is blue 1332 PB2W1PW1 PB2BIPBI l 32 2 1 2 WIN Probability that the first sock is white given the second sock is blue P32W1PW1 Bayes39 rule PW B 1 2 PltBgW1gtPltW1gt PBgW1PW1 PBgBlPBl 9 2g awe 3 Discrete probability distributions Random variable a numerical outcome of a random experiment usually denoted with an upper case letter e g Y ex of blue socks SAT of a randomly drawn student democrats in a random sample of voters l day39s growth dry weight of a plant Probability distribution collection of all possible outcomes of a random variable and their associated probabilities a particular outcome usually denoted with a lower case letter eg y Discrete probability distribution random variable has a nite or countably in nite number of states 1 Binomial distribution Y a random variable is the number of successes in n independent identical trials in which each trial can be a success or a failure possible outcomes are y O 1 2 3 n For each trial the probability of success is 7r 0 lt 7r lt 1 not the pi from a circle The binomial distribution has probabilities given by n mm W P0 y Py for y O 1 2 3 n These probabilities add to 1 PO P1 P2 Pn 1 The expected value or mean of the binomial random variable Y EYMOPOlP1nPn ALA Variance of the random variable Y WY 02 0 M2P0 1 Hypo n u2Pn EA n7rl 7T Common notation Y N binomialn 7r Y has a binomial distribution with parameters n and 7r SAS The function RANBINseed n p returns a binomial random variable with trials 11 and success probability p set seed O and SAS will use the computer clock time as seed The function PROBBNMLp n X computes the probability that an observation from a binomia1n p distribution will be less than or equal to X exercise concept of mean and variance of a discrete distribution suppose Y has a rectangular distribution given by y123456 distribution of the result of rolling a die a Draw a picture of the probability distribution b Calculate the expected value of Y c Calculate the variance of Y 2 Poisson distribution e I LMy y0123 Here 6 271828 and u is a parameter u gt O This is a distribution with positive probability on all the nonnegative integers PO P1 P2 fay 1 Poisson distribution arises as a model of rare events radioactive decays in a unit of time incoming cosmic rays in a unit of time plant stems in a sample plot steelhead caught in 1 hr crimes reported in Moscow ID in 1 day car accidents reported in a stretch of U895 in 1 week Note that zero is a possible outcome in the Poisson distribution Some other properties E0 M VY 02 u Variance equals the mean in the Poisson distribution Notation Y N Poissonu True fact suppose Y N binomialn 7r that is 7Tyl 7rny Suppose n is large and 7r is small Then 6 y where u 2 mr Poisson approximation to the binomial SAS RANDPOIseed m generates a Poisson random variable with mean m POISSONm X calculates PY S X where Y N Poissonu 3 Multinomial distribution a multivariate distribution ktypes O O V V samplen O V O with replacement VVVV 7T1 proportion of type 1 in the urn g 2 proportion of type 2 V in the urn 7rk propostion of type k V in the urn The 7rj39s are constants parameters 7T17T27Tk 1 Y1 Y2 Yk random variables Y1 number of type 1 in the sample Y2 number of type 2 V in the sample Yk number of type k V in the sample Y1 Y2quot Yk n The Yj39s are dependent the value of one affects the others PY1 y1 and Y2 y2 and and Yk yk n Tryl y2 yk y1y239 yk 17T2 quotWk where y1 y2 31 are any nonnegatiVe integers that add to n all possible outcomes examples Y1 democrats Y2 republicans Y3 greens Y4 other in a random sample of n voters Y1 genotype AA BB Y2 AA Bb Y3 AA bb Y4 genotype Aa BB Y9 genotype aa bb in a random sample of n people A population has n mice Catch mice by livetrapping on two sampling occasions uniquely identify each one caught Y1 mice caught in the rst sample as well as the second Y2 mice caught in the rst sample but not in the second Y3 mice not caught in the rst sample but caught in the second Y4 mice not captured in either sample unobserved n unknown

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