Itroductio to Probability ad Statistics Twelfth Editio Robert J. Beaver Barbara M. Beaver William Medehall Presetatio desiged ad writte by: Barbara M. Beaver
Itroductio to Probability ad Statistics Twelfth Editio Chapter 5 Several Useful Discrete Distributios Some graphic scree captures from Seeig Statistics Some images 2001-(curret year) www.arttoday.com
Itroductio Discrete radom variables take o oly a fiite or coutably umber of values. Three discrete probability distributios serve as models for a large umber of practical applicatios: The biomial radom variable The Poisso radom variable The hypergeometric radom variable
The Biomial Radom Variable The coi-tossig experimet is a simple example of a biomial radom variable. Toss a fair coi = 3 times ad record x = umber of heads. x p(x) 0 1/8 1 3/8 2 3/8 3 1/8
The Biomial Radom Variable May situatios i real life resemble the coi toss, but the coi is ot ecessarily fair, so that P(H) 1/2. Example: A geeticist samples 10 people ad couts the umber who have a gee liked to Alzheimer s disease. Coi: Head: Tail: Perso Has gee Does t have gee umber of tosses: P(H): = 10 P(has gee) = proportio i the populatio who have the gee.
The Biomial Experimet 1. The experimet cosists of idetical trials. 2. Each trial results i oe of two outcomes, success (S) or failure (F). 3. The probability of success o a sigle trial is p ad remais costat from trial to trial. The probability of failure is q = 1 p. 4. The trials are idepedet. 5. We are iterested i x, the umber of successes i trials.
Biomial or ot? Very few real life applicatios satisfy these requiremets exactly. Select two people from the U.S. populatio, ad suppose that 15% of the populatio has the Alzheimer s gee. For the first perso, p = P(gee) =.15 For the secod perso, p P(gee) =.15, eve though oe perso has bee removed from the populatio.
The Biomial Probability Distributio For a biomial experimet with trials ad probability p of success o a give trial, the probability of k successes i trials is P( x k) C k p k q k! k!( k)! p k q k for k 0,1,2,.... Recall C k! k!( k)! with! ( 1)( 2)...(2)1 ad0! 1.
The Mea ad Stadard Deviatio For a biomial experimet with trials ad probability p of success o a give trial, the measures of ceter ad spread are: Mea: Variace: p pq Stadarddeviatio : 2 pq
MY APPLET Example A marksma hits a target 80% of the time. He fires five shots at the target. What is the probability that exactly 3 shots hit the target? = 5 success = hit p =.8 x = P( x 3) C 3 p 3 q 3 5! 3 53 (.8) 3!2! (.2) # of hits 10(.8) 3 (.2) 2.2048
MY APPLET Example What is the probability that more tha 3 shots hit the target? P( x 3) C 5 4 p 4 q 5 4 C 5 5 p 5 q 55 5! (.8) 41!! 4 (.2) 1 5! (.8) 5!0! 5 (.2) 0 5(.8) 4 (.2) (.8) 5.7373
Cumulative Probability Tables You ca use the cumulative probability tables to fid probabilities for selected biomial distributios. Fid the table for the correct value of. Fid the colum for the correct value of p. The row marked k gives the cumulative probability, P(x k) = P(x = 0) + + P(x = k)
MY APPLET Example k p =.80 0.000 1.007 2.058 3.263 4.672 5 1.000 What is the probability that exactly 3 shots hit the target? P(x = 3) = P(x 3) P(x 2) =.263 -.058 =.205 Check from formula: P(x = 3) =.2048
MY APPLET Example k p =.80 0.000 1.007 2.058 3.263 4.672 5 1.000 What is the probability that more tha 3 shots hit the target? P(x > 3) = 1 - P(x 3) = 1 -.263 =.737 Check from formula: P(x > 3) =.7373
Example Here is the probability distributio for x = umber of hits. What are the mea ad stadard deviatio for x? Mea : p 5(.8) 4 Stadarddeviatio: pq 5(.8)(.2).89
Example A studet takes a multiple-choice quiz that cosists of five questios. Each questio has four possible aswers, oly oe of which is correct. Suppose that the studet was ot study for the quiz ad just guesses at each questio. Fid (a) the probability that the studet gets exactly three questios correct. (b) the probability that the studet gets at least three questios correct. (c) the expected score for a studet. (d) the variability of the score for a studet.
The Poisso Radom Variable The Poisso radom variable x is a model for data that represet the umber of occurreces of a specified evet i a give uit of time or space. Examples: The umber of calls received by a switchboard durig a give period of time. The umber of machie breakdows i a day The umber of traffic accidets at a give itersectio durig a give time period.
The Poisso Probability Distributio x is the umber of evets that occur i a period of time or space durig which a average of such evets ca be expected to occur. The probability of k occurreces of this evet is For values of k = 0, 1, 2, The mea ad stadard deviatio of the Poisso radom variable are Mea: P( x k) Stadard deviatio: k e k!
Example The average umber of traffic accidets o a certai sectio of highway is two per week. Fid the probability of exactly oe accidet durig a oe-week period. P( x 1 k e 2 e 1) 2e 2. 2707 k! 1! 2
Exercise The average umber of traffic accidets o a certai sectio of highway is two per week. Fid the probability of at most three accidets o this sectio of highway durig a 2-week period.
Cumulative Probability Tables You ca use the cumulative probability tables to fid probabilities for selected Poisso distributios. Fid the colum for the correct value of. The row marked k gives the cumulative probability, P(x k) = P(x = 0) + + P(x = k)
Example k = 2 0.135 1.406 2.677 3.857 4.947 5.983 6.995 7.999 8 1.000 What is the probability that there is exactly 1 accidet? P(x = 1) = P(x 1) P(x 0) =.406 -.135 =.271 Check from formula: P(x = 1) =.2707
Example k = 2 0.135 1.406 2.677 3.857 4.947 5.983 6.995 7.999 8 1.000 What is the probability that 8 or more accidets happe? P(x 8) = 1 - P(x < 8) = 1 P(x 7) = 1 -.999 =.001 This would be very uusual (small probability) sice x = 8 lies x 8 2 z 1.414 4.24 stadard deviatios above the mea.
Exercise
The Hypergeometric m m m m m m m Probability Distributio The M&M problems from Chapter 4 are modeled by the hypergeometric distributio. A bowl cotais M red cadies ad -M blue cadies. Select cadies from the bowl ad record x the umber of red cadies selected. Defie a red M&M to be a success. The probability of exactly k successes i trials is P( x k) C M k C C M k
The Mea ad Variace The mea ad variace of the hypergeometric radom variable x resemble the mea ad variace of the biomial radom variable: 1 : Variace Mea: 2 M M M m m m m m m m
Example A package of 8 AA batteries cotais 2 batteries that are defective. A studet radomly selects four batteries ad replaces the batteries i his calculator. What is the probability that all four batteries work? Success = workig battery = 8 P( x 6(5) / 4) 2(1) 6 C4C C M = 6 = 4 8(7)(6)(5) / 4(3)(2)(1) 70 8 4 2 0 15
3 8 6 4 M Example What are the mea ad variace for the umber of batteries that work?.4286 7 4 8 2 8 6 4 1 2 M M
Exercise A cady dish cotais five blue ad three red cadies. A child selects three cadies without lookig. What is the probability that (a) there are two blue ad oe red cadies i the selectio? (b) the cadies are all red? (c) the cadies are all blue?
Key Cocepts I. The Biomial Radom Variable 1. Five characteristics: idetical idepedet trials, each resultig i either success S or failure F; probability of success is p ad remais costat from trial to trial; ad x is the umber of successes i trials. 2. Calculatig biomial probabilities a. Formula: P( x k) b. Cumulative biomial tables k c. Idividual ad cumulative probabilities usig Miitab 3. Mea of the biomial radom variable: p 4. Variace ad stadard deviatio: 2 pq ad C k p k q pq
Key Cocepts II. The Poisso Radom Variable 1. The umber of evets that occur i a period of time or space, durig which a average of such evets are expected to occur 2. Calculatig Poisso probabilities a. Formula: b. Cumulative Poisso tables P( x k) c. Idividual ad cumulative probabilities usig Miitab 3. Mea of the Poisso radom variable: E(x) 4. Variace ad stadard deviatio: 2 ad k e k! 5. Biomial probabilities ca be approximated with Poisso probabilities whe p < 7, usig p.
Key Cocepts III. The Hypergeometric Radom Variable 1. The umber of successes i a sample of size from a fiite populatio cotaiig M successes ad M failures 2. Formula for the probability of k successes i trials: 3. Mea of the hypergeometric radom variable: 4. Variace ad stadard deviatio: M 1 2 M M M k M k C C C k x P ) (