Binomial Random Variables. Binomial Distribution. Examples of Binomial Random Variables. Binomial Random Variables

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1 Binomial Random Variables Binomial Distribution Dr. Tom Ilvento FREC 8 In many cases the resonses to an exeriment are dichotomous Yes/No Alive/Dead Suort/Don t Suort Binomial Random Variables When our focus is conducting an exeriment n times indeendently and observing the number x of times that one of the two outcomes occurs This x is a Binomial Random Variable We can exloit this by using known formulas for a robability distribution Examles of Binomial Random Variables, eole are olled in a telehone survey and asked if they suort George W. Bush The resonses are Yes () or No () The roortion saying yes is designated as (-) is the roortion saying No Binomial Random Variable Yes it is a binomial random variable Conduct an exeriment, times and observe the number x of times that Yes occurs Characteristics of a Binomial Random Variable The exeriment consists of n identical trials There are only two outcomes on each trial. Outcomes can be denoted as S for Success F for Failure

2 Characteristics of a Binomial Random Variable (cont.) The robability of S (success) remains the same from trial to trail Denoted as the roortion The robability of F (failure) Denoted as =(-) The trials are indeendent of each other The binomial random variable x is the number of Successes in n trials Also refer to Conditions Reuired for a Binomial Exeriment on P6 of book. Examle : Marketing examle Marketing survey of randomly chosen consumers Record their references for a new and an old diet soda ask them to choose their reference Let x be number of who choose the new brand This is a binomial random variable Conduct an exeriment times and observe the number x of times that Yes occurs Fitness Examle Heart Association says only % of adults over can ass the fitness test Suose eole over are selected at random Let x be the number who ass the minimum reuirements Find the robability distribution for x Conduct an exeriment times and observe the number x of times that ass occurs How to solve the fitness roblem the way we used with discrete random variables. List the events. List the samle oints that refer to that event. Calculate the robabilities =. and = (. -.) =.9 Event x Samle Points FFFF (.9)(.9)(.9)(.9) =.66 I multily through on the robabilities because Each trial is indeendent of the others Solve for Each Event Event x SFFF FSFF Pass Fail FFSF FFFS SSFF SFSF Pass Fail SFFS FSSF FSFS FFSS SSSF SFSS Pass Fail SSFS FSSS Pass FFFF Notation SSSS (.9)(.9)(.9)(.9) =.66 [(.)(.9) ] =.96 6[(.) (.9) ] =.86 [(.) (.9)] =.6 (.)(.)(.)(.) =. Fitness Examle When x = P =.66 Distribiution of X When x = One Pass P = When x = Two ass P =.86.. When x= Three. ass P =.6. When x= Four ass P =..

3 Fitness Examle Find the robability that none of the adults ass the test P(x=) =.66 Find the robability that of adults ass the test P(x=) =.6 When we have many trials the formulas get comlicated We can also use the binomial robability distribution formula Using factorial notation = n(n-)(n-) (n-(n-))! = xxxx =! =,!=,!=x=, The formula for any x in n trials is: n! Binomial Distribution Formula (P8) n! aka = n x x n x Most calculators will do all or art of this become familiar before trying it out Note: it uses the Combinatorial Rule as the first art of the formula What defines a binomial robability distribution? = of a success on a single trial = (-) robability of failure n= number of trials x = number of successes in n trials n! For x= in the fitness examle! ) = (.)!( )! (.9) = (.)(.9) ( )() = (.9) =.6 6 n! This matches the number we generated the other way Fitness Examle Table Event x SSFF SFSF Pass Fail SFFS FSSF FSFS FFSS SSSF SFSS Pass Fail SSFS FSSS Pass Notation FFFF SFFF FSFF Pass Fail FFSF FFFS SSSS

4 For x= in fitness examle! ) = (.)!( )! (.9) = (.)(.8) ( )( ) = (.8) =.86 n! This matches the number we generated the other way Fitness Examle Table Event x SFFF FSFF Pass Fail FFSF FFFS SSFF SFSF Pass Fail SFFS FSSF FSFS FFSS SSSF SFSS Pass Fail SSFS FSSS Pass Notation FFFF SSSS Mean of a Binomial Random Variable Since a binomial is only a dichotomy, the formulas for the mean and the standard deviation will simlify From := xa To := n (P) Variance and Standard Deviation of a Binomial Random Variable From F = (x-:) To F = n (P) The standard deviation is then σ = n Mean and Standard Deviation for Fitness Examle Heart Association says only % of adults over can ass the fitness test Thus the roortion assing was estimated at., and n for the roblem was eole := n = (.) =. F = n = (.)(.9) =.6 F =.6 Fitness Examle Table for mean and variance Event x Notation FFFF SFFF FSFF Pass Fail FFSF FFFS SSFF SFSF Pass Fail SFFS FSSF FSFS FFSS SSSF SFSS Pass Fail SSFS FSSS Pass SSSS

5 I could have solved for the mean using the formula for discrete random variables To solve for the mean I would have: n E( = xi x i ) = µ i= E( = ()(.66) + ()(.96) + ()(.86) + ()(.6) + ()(.) E( =. Binomial aroach E( = n = (.) =. I could have solved for the Variance using the formula for discrete random variables To solve for the variance I would have: E n [( x µ ) ] = ( xi µ ) x i ) = σ i= E(x-:) = ( -.) (.66) + (-.) (.96) + (-.) (.86) + (-.) (.6) + (-.) (.) E(x-:) =.6 Binomial aroach E( = n = (.)(.9) =.6 Nitrous Oxide Examle Suose we were recording the number of dentists that use nitrous oxide (laughing gas) in their ractice We know that 6% of dentists use the gas. =.6 and =. Let X = number of dentists in a random samle of five dentists use use laughing gas. n = Nitrous Oxide Examle We said the robability that a dentist uses nitrous oxide is.6 How would you assign robabilities to the values x could take when we randomly select five dentists? X Solve for Each Event Event x FFFFF SFFFF FSFFF FFSFF FFFSF Pass Fail FFFFS Pass Fail There Notation SSFFF SFSFF SFFSF SFFFS FSSFF FSFSF FSFFS FFSSF FFSFS FFFSS Is (.)(.)(.)(.)(.) =. [(.6)(.)(.)(.)(.)] x =.768 [(.6)(.6)(.)(.)(.)] x =. More!! X P(X) Nitrous Oxide Examle So what other way can we get to the robabilities? n!

6 Nitrous Oxide Examle Solve for x=)! ) = (.6) (.)!( )! () = (.6)(.6) = 6() n!.6 X).... Distribution of the Discrete Variable X Distribution of X Number of Dentists E(X) = F =.998 F =.9 X).... Distribution of the Discrete Variable X Distribution of X Number of Dentists E(X) = F =.998 F =.9 E(X) = n=(.6) = F = n = (.6)(.) =. F =.9 Examle: Seedling Survival An agronomist knows from ast exerience that 8% of a citrus variety seedling will survive being translanted. If we take a random samle of 6 seedlings from current stock, what is the robability that exactly seedlings will survive? Examle: Seedling Survival For the roblem we can calculate =.8 =. : = n = 6(.8) =.8 F = n = 6(.8)(.) =.96 F =.98 Examle: Seedling Survival that exactly survive is 6! 6 ) = (.8) (.)!(6 )! 6 = (.6)(.6) ( )( ) 7 = (.) =.6 8 6

7 Examle: Seedling Survival Solve for Each Event that exactly survive is 6! 6 ) = (.8) (.)!(6 )! 6 = (.)(.8) ( )( ) 7 = (.) =.8 6 Event x = All fail x = One ass x = Two ass x = Three ass x = Four ass x = Five ass x = 6 Six ass Look at the Cumulative Probabilities Event x = All fail x = One ass x = Two ass x = Three ass x = Four ass x = Five ass x = 6 Six ass Cumulative Citrus Examle Mean = 6(.8) =.8 Std Dev =.98 It Makes Sense! Our exectation is that most seedlings will survive (i.e..8 of 6) Look at the cumulative robability.... Distribiution for Citrus Examle 6 Move to the Binomial Table Binomial Table n=6 We can also use a table to hel Aendix B, Table B (Page 7) contains cumulative robabilities for n=, 6, 7, 8, 9,,,, and Each table lists values of across the to P =.,.,.,.,.,,.9,.99 k = # of successes k P

8 Binomial Table NOTE: The table is cumulative binomial robabilities, cumulative u to an including the value for k This means to find exact robabilities you might have to subtract two table values Binomial Probabilities Using the Table for Citrus Examle We said the robability that survive is.6 From the Table Cumulative u to is. Subtract the robabilities for u to (.99) =.6 You have to be careful using the Table! Binomial Formula using Excel In Excel, the formula for the Binomial Distribution function is: BINOMDIST(X,N,P,cumulative) X is the number of successes N is the number of indeendent trials P is the robability of success on each trial Cumulative is an argument Entering TRUE gives a cumulative robability u to and including X successes Entering FALSE gives the exact robability of X successes in N trials Binomial Formula using Excel For our examle of citrus lants BINOMDIST(,6,.8,TRUE) cumulative robability u to and including successes.696 BINOMDIST(,6,.8,FALSE) the exact robability of X successes in N trials =.6 Look at the Citrus Seedling Table Event x = All fail x = One ass x = Two ass x = Three ass x = Four ass x = Five ass x = 6 Six ass Cumulative Excel Binomial Distribution File =.8 =. X X) Cum X) n = Mean Variance Std Dev #NUM! #NUM! 8 #NUM! #NUM! 9 #NUM! #NUM! #NUM! #NUM! Formula Cum = FALSE.6 X successes = Formula Cum= TRUE.696 8

9 The Rare Event Aroach What if we had 6 seedlings selected randomly and all of them died? Given =.8, this would be a very rare event P(x=) =. Was this just by chance???? Examle Problem A study in the American Journal of Public Health found that 8% of female Jaanese students from heavy-smoking families showed signs of nasal allergies Consider a random samle of female Jaanese students exosed daily to heavy smoking What is the robability that fewer than of the students will have nasal allergies? Answer to Problem What is the robability that more than of the students will have nasal allergies? Let s revisit the sychic roblem Let s view this as a discrete random variable a binomial random variable Remember that a crystal is randomly laced under one of ten boxes and the sychic is asked to guess where it is. This exeriment is reeated seven times, and x is the number of correct decisions in seven tries. Thus it is a Binomial random variable. If the sychic is guessing, what is the value of, the robability of a correct decision on each trial? =. X the robability of a success is. Can you fill in the rest of the table? 9

10 To solve Use the table on age 7 n = 7 =. k = the values of our discrete random variable For x=) For k =, the robability is.8 which is the cumulative robability u and including To find the exact x=), subtract the value for k= from the value k = x=) = =.7 To solve Use the formula 7! ) = (.) (.9)!(7 )! 7 = (.) =.7 7 Solve for all Solve for Exected Value and Variance X 6 7 X Exected value = mean = n = 7*. =.7 Variance = n = 7*(.)*(.9) =.6 Standard Deviation = (.6). =.79 Can you solve for the mean and standard deviation of this binomial random variable? Poisson Distribution Alies to situations where we describe the number of events occurring in a secific time eriod or in a secific area x λ e = x! λ Where 8 = : e = natural logarithm =.78