Chapter 9 Binomial population distribution 9.1 Definition of a Binomial distributio If the random variable has a Binomial population distributio i.e., then its probability function is given by p n n ( p (1 for 0, 1,, 3,..., n Binomial distribution is often a good model for discrete random variables which are counts of the number of times an event occurs out of a total number of n trials or repetitions of an eperiment, e.g. = number of successes from n trials = number of heads from n tosses of a coin = number of sies from n tosses of a dice = number of people of type from a random sample of size n = number of Labour supporters in a random sample of n people = number of girls born from a random sample of n babies = number of people cured by a treatment from a random sample of size n = number of insects killed by a dose of insecticide from a random sample of size n NOTE the range of possible values of the variable is 0, 1,, 3,..., n, i.e. is a discrete variable which takes non-negative integer values from 0 to n inclusive. R.. Rigby and D. M. Stasinopoulos September 005 73
Eamples of binomial distributions Figure.1: Binomial distribution with n = 10 and p = 0.1 Figure.: Binomial distribution with n = 10 and p = 0.5 Figure.3: Binomial distribution with n = 10 and p = 0.7. R.. Rigby and D. M. Stasinopoulos September 005 74
9. Theoretical conditions leading to a Binomial variable Theorem 1 If counts the number of times event occurs out of n trials (i.e. repetitions of an eperiment, then has a Binomial distributio with p n n ( p (1 for 0, 1,, 3,..., n provided i the outcomes of the trials are independent of each other ii the probability of event occurring is the same value p for each of the n trials Proof 1 First consider n=3 = number of s from 3 repetitions of the eperiment R.. Rigby and D. M. Stasinopoulos September 005 75
e.g. = number of heads from 3 tosses of a coin where p = probability of heads on each toss of the coin and q = (1- = probability of tails on each toss of the coin sample = outcomes HHH 3 3 p 3 HHT HTH 3p q HTT 1 THH THT 1 1 3pq TTH 1 TTT 0 0 q 3 Proof 1 Now consider the general case of n trials (e.g. n tosses of a coin = = (number of different ways to get heads from n tosses*(probability of each way (number of different ways to get heads from n tosses = (number of ways to choose from n with NO repetition and order NOT important = C n (probability of each way = HH..H TT.T = pp.pqq.q = p q n- = p ( 1p n- n- Hence n n p (1 R.. Rigby and D. M. Stasinopoulos September 005 7
9.3 Population summary measures for a Binomial variable mean np variance npq n 1p standard deviation npq 9.4 Procedure for finding a Binomial probability 1 identify the event, success find the probability p of event occurring in each of the trials 3 identify the total number n of trials n n p (1 9.5 Eamples Eample 1 Toss a fair coin 10 times and count the number of heads from the 10 tosses. What is the distribution of? B (10,0.5 What is the probability of getting heads from 10 tosses? 10 10 p (1 p 45*0.5 *0.5 45* 0.5 0.0439 i.e. 4.39% What is the probability of getting 10 heads from 10 tosses? 10 10 p (1 p 45*0.5 *0.5 45* 0.5 0.0439 i.e. 0.0977% R.. Rigby and D. M. Stasinopoulos September 005 77
Eample Toss a fair dice 0 times and count the number of sies from the 0 tosses. What is the distribution of? B (0, 1 What is the probability of getting 5 sies from 0 tosses? 0 5 15 1 5 5 5 p (1 15504* * 0.194 i.e. 1.94% What is the probability of getting 0 sies from 0 tosses? 0 0 0 0 1 5 0 0 p (1 1* * 0.01 i.e..1% Eample 3 The probability a newborn baby is a girl is 0.47 (Statistical bstract of the United States Let count the number of girls out of babies born in a hospital in the United States. What is the distribution of? B (,0.47 5 0 15 What is the probability of getting girls from babies? 0.093 p (1 p * 0.47 * (1 0.47 *0.47 * 0.513 What is the probability of getting or more girls from the babies? p ( 7 p ( = 0.093 7 1 7 1 7 7 p (1 * 0.47 *0.513 0.07 0 0 p (1 p 1* 0.47 * 0.513 0.003 p ( p ( 7 = 0.093 + 0.07 + 0.003 = 0.191 R.. Rigby and D. M. Stasinopoulos September 005 7
9. Using MINITB to calculate Binomial probabilities In eample 3 above, counts the number of girls from babies born in a hospital in the US B (,0.47 Use MINITB to output the values of the pf and cdf for the B (,0.47 model Type values 0, 1,, 3,, into C1 Probability Function > Calc > Random Distributions > Binomial Probability Number of trials Probability 0.47 Input Variable C1 OK Binomial with n = and p = 0.47000 P( = 0.00 0.004 1.00 0.034.00 0.110 3.00 0.9 4.00 0.77 5.00 0.071.00 0.093 7.00 0.07.00 0.003 Cumulative Distribution Function > Calc > Random Distributions > Binomial Cumulative Prob Number of trials Probability 0.47 Input Variable C1 OK Binomial with n = and p = 0.47000 P( <= 0.00 0.004 1.00 0.041.00 0.13 3.00 0.391 4.00 0.4 5.00 0.719.00 0.970 7.00 0.99.00 1.0000 R.. Rigby and D. M. Stasinopoulos September 005 79
9.7 Using MINITB to calculate and plot Binomial probabilities and cumulative probabilities C1 C C3 = <= 0 0.004797 0.0040 1 0.034 0.0413 0.1103 0.1 3 0.90 0.3907 4 0.799 0.477 5 0.0710 0.717 0.09303 0.97017 7 0.03 0.994 0.00314 1.00000 0.3 0. = 0.1 0.0 0 1 3 4 5 7 1.0 <= 0.5 0.0 0 1 3 4 5 7 R.. Rigby and D. M. Stasinopoulos September 005 0
9. Normal approimation to the Binomial If has a Binomial distributio then can be approimated by a Normal distribution with the same mean and variance as the Binomial, i.e. ~ N (, N ( np, npq where q = 1 - p provided n is large ( n 0 and p is not too etreme i.e. not too close to 0 or 1 ( 0.1 0. 9 p Eample 0,0.3 ~ N (0 *0.3,0* 0.3*0.7 N (1,1. R.. Rigby and D. M. Stasinopoulos September 005 1
PRCTICL Discrete Distributions Q1 Binomial random variables ccording to the US National Institute of Mental Health 0% of adult mericans suffer from a psychiatric disorder. [Weiss, p309] Let count the number of adult mericans suffering from a psychiatric disorder from a random sample of 0. i state the distribution of ii state the mean and standard deviation of iii use MINITB to calculate in C and plot the probability function for by typing the values 0, 1,, 3,, 0 into C1 and then > Calc > Probability Distributions > Binomial Probability Number of trials n Probability p Input Variable C1 OK > Calc > Probability Distributions > Binomial Output Variable C OK > Graph > Scatterplot > Simple Y C C1 > Data View Data Display Project Lines OK > OK iv use MINITB to calculate in C3 and then plot the cumulative distribution function for, using for the plot > Graph > Scatterplot > With Connect Line Y C3 C1 > Data View Data Display Symbols onnect Line OK > OK Edit the graph by double clicking on the curve > Options Step > Left OK v use the output from the cdf of from iv to find p ( 3, p ( 3, p (, p (, p ( 4, p ( 4, 3 R.. Rigby and D. M. Stasinopoulos September 005
Q Binomial random variables In the UK population % of males but only 0.4% of females suffer from colour blindness. Random samples of males and females each of size 100 were obtained and the numbers suffering colour blindness was recorded as variables 1 and respectively i state the distributions of 1 and ii state the mean and standard deviation of 1 and iii use MINITB to calculate the following p ( 1 10, 10 i use MINITB to find the median and quartiles of 1 Q3 Poisson random variables ~ P( The incidence of major earthquakes follows a Poisson process with mean number of earthquakes per year equal to 035. [and hence times between earthquakes have an Eponential distribution with mean 1 1 0. 35 1197. yrs (= 437 days Hand et al., p03] Let count the number of earthquakes in a year. i state the distribution of ii state the mean and standard deviation of iii use MINITB to calculate the following p (, p (, 0 Let count the number of earthquakes in a five-year period, then it can be proved that ~ P(5 i state the mean and standard deviation of ii use MINITB to calculate the following p ( 5, 0 R.. Rigby and D. M. Stasinopoulos September 005 3