STAT371 DISCUSSION 4 September 29 2002 TA Ruiyan Luo Office 4268 CSSC Office hours TW 2 30 3 30pm Phone number 262 8182 E mail rluo stat wisc edu 1 Bionomial distribution Four conditions for a binomial random varialbe Binary outcomes there are two possible outcomes for each trial success and failure Independent rials the outcomes of the trials are independent of each other n is fixed the number of trials n is fixed in advance Same value of p the probability of a success on a single trial is the same for all trials The binomial distribution formula For a binomial random variable the probability that the n trials result in j successes and n j failures is given by the following formula P r jsuccesses n Cj pj 1 p n j n and x x x 1 x 2 2 1 0 1 where n Cj j n j C has some properties n j n C0 n Cn 1 n Cj n Cn j Properties of binomial distribution expectation or mean of X is np q variance is np 1 p standard deviation is np 1 p the binomial distribution is symmetric if and only if p 0 5 1 2 Normal distribution If Y follows a normal distribution with mean and standard deviation then it is common to write Y N Its density function is 1 y 2 1 f y e 2 2 By standardization formula Z Y random variable Z has density function z2 1 f z e 2 2 which is called standard normal distribution with mean 0 and standard deviation 1 Pr Z is between a and b area under the standard normal curve between a and b The tabel in the book gives the area under the normal curve below a spedified value of z Pr Z z area to the left of z given in table Pr Z z area to the right of z 1 area to the left of z Pr a Z b area to the left of b area to the left of a Given a probability from the normal table we can get Z such that P r Z Z 1 then Y Z satisfying that P r Y Y 1 2
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