Binomial population distribution X ~ B(
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1 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
2 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
3 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
4 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
5 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? p (1 p 45*0.5 *0.5 45* i.e. 4.39% What is the probability of getting 10 heads from 10 tosses? p (1 p 45*0.5 *0.5 45* i.e % R.. Rigby and D. M. Stasinopoulos September
6 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? p ( * * i.e. 1.94% What is the probability of getting 0 sies from 0 tosses? 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 (, What is the probability of getting girls from babies? p (1 p * 0.47 * ( *0.47 * What is the probability of getting or more girls from the babies? p ( 7 p ( = p (1 * 0.47 * p (1 p 1* 0.47 * p ( p ( 7 = = R.. Rigby and D. M. Stasinopoulos September 005 7
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 = P( = Cumulative Distribution Function > Calc > Random Distributions > Binomial Cumulative Prob Number of trials Probability 0.47 Input Variable C1 OK Binomial with n = and p = P( <= R.. Rigby and D. M. Stasinopoulos September
8 9.7 Using MINITB to calculate and plot Binomial probabilities and cumulative probabilities C1 C C3 = <= = <= R.. Rigby and D. M. Stasinopoulos September 005 0
9 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 ( 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
10 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
11 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 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
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