BINOMIAL DISTRIBUTIONS 1 10 The sample count random variable X In these trials the sample size n will be the same for all the samples The population proportion of successes p is considered to be a fixed quantity but the sample count of successes x will differ from sample to sample Example randomly select samples of 4 persons each and count how many in each sample were born in Florida Being born in Florida would be a success being born anywhere outside of Florida would be a failure The possible counts would be 0 1 2 3 4 Sample Space of the random variable X Random Process number of successes x X POPULATION proportion of successes p SAMPLING DISTRIBUTION OF COUNT X sample size n success probability p Sampling distribution The distribution of all possible values of a specific statistic obtained from all possible simple random samples of the same size drawn from the same population Next Step What is the PMF of this random variable 2022 Radha Bose Florida State University Department of Statistics BINOMIAL DISTRIBUTIONS 2 10 RATIONALE BEHIND THE PMF F a randomly selected FSU student was born in Florida N FC a randomly selected FSU student was not born in Florida Let us suppose that P F 0 62 P N 0 38 X the number who were born in Florida among 4 randomly selected FSU students Values of X Possible arrangements 0 1 2 3 4 N N N N F N N N N F N N N N F N N N N F F F N N F N F N F N N F N F F N N F N F N N F F F F F N F F N F F N F F N F F F F F F F Probability for each value of X P X 0 1 0 38 0 38 0 38 0 38 0 384 1 0 620 0 384 P X 1 4 0 62 0 38 0 38 0 38 4 0 621 0 383 P X 2 6 0 62 0 62 0 38 0 38 6 0 622 0 382 P X 3 4 0 62 0 62 0 62 0 38 4 0 623 0 381 P X 4 1 0 62 0 62 0 62 0 62 0 624 1 0 624 0 380 Total 1 P X r of arrangements success probabilityvalue of X failure probabilitynumber of remaining spots nCr pr qn r 2022 Radha Bose Florida State University Department of Statistics 3 10 BINOMIAL DISTRIBUTIONS The Family of Bernoulli Random Variables or Bernoulli Distributions Bernoulli p X a randomly selected value from the sample space S 0 1 Parameter is success probability p where p PMF P X 1 p P X 0 q 1 p E X p X pq Shape of probability histogram perfectly symmetric and uniform when p 0 5 right skewed when p 0 5 left skewed when p 0 5 The Family of Binomial Random Variables or Binomial Distributions Bin n p X a randomly selected value from the sample space S 0 1 2 n Parameters are sample size n and 0 success probability p where p 1 q 1 p failure probability PMF P X r nCrprqn r r 0 1 S E X X np X npq Shape of probability histogram perfectly symmetric and bell shaped when p 0 5 right skewed when p 0 5 left skewed when p 0 5 skewness increases as p moves away from 0 5 tail probabilities decrease as n gets larger tail area shrinks If X Bin n p and Y Bin m p are independent with the same success probability then X Y Bin n m p X Bin n p the sum of n Bernoulli p random variables FYI the Binomial Coefficient nCr nCr n r n r n x n 1 x n 2 x x 3 x 2 x 1 r x r 1 x x 2 x 1 n r x n r 1 x x 2 x 1 nCr is the number of ways of choosing r objects out of n objects when ordering does not matter or the number of ways of arranging n objects when r of them are of one type and n r are of another type It is the rth entry from the left in the nth row from the top of Pascal s Triangle where the numbering of the rows and the entries starts at zero 2022 Radha Bose Florida State University Department of Statistics Pascal s Triangle 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 etc BINOMIAL DISTRIBUTIONS 4 10 Quick Tips nC0 nCn 1 nC1 nCn 1 n nCr nCn r REAL LIFE APPLICATION BINOMIAL SETTING X the number of subjects who possess a certain characteristic in a sample of n subjects a fixed number of trials n each subject either possesses the characteristic success or they don t possess the characteristic failure just two possible outcomes at each trial the subjects possess or not the characteristic independently of each other independent outcomes across trials p and q are the same for all subjects constant probabilities When the above four conditions are present then it is a Binomial setting the sampling distribution of the count X will be a Binomial distribution and X will be a Binomial random variable INTERPRETING IN CONTEXT Binomial Probability P X r There is a insert here the probability you are interpreting probability that r subjects will possess the success characteristic among n randomly selected subjects Insert here the percentage you are interpreting of all possible random samples of n subjects each are expected to have r subjects in each sample who possess the success characteristic The above sentences can be catered for all other cases such as when X r or X r etc Binomial Mean If we consider all possible random samples of n subjects each and we count how many possess the success characteristic in each sample then the average of those counts is expected to be Binomial Standard Deviation If we consider all possible random samples of n subjects each and we count how many possess the success characteristic in each sample then the standard deviation of those counts is expected to be 2022 Radha Bose Florida State University Department of Statistics BINOMIAL DISTRIBUTIONS 5 10 Example Data from STA 2171 0001 SuC22 collected on May 9th 2022 Based on a survey conducted in class we estimate that roughly 11 of all FSU students normally use Discover card Let X the number who normally use Discover card among 200 randomly selected FSU students a We can consider X to be Binomially distributed because there are a fixed number of trials n 200 each trial consists of randomly selecting a an fsu student there are only two possible outcomes use discover card and dont use discover card the probability of success is the same for all trials p 11 trials can be considered independent because …