Discrete and Continuous Probability Distributions.ppt

1、Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-1,Business Statistics: A Decision-Making Approach 6th Edition,Chapter 5Discrete and Continuous Probability Distributions,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-2,Chapter G

2、oals,After completing this chapter, you should be able to: Apply the binomial distribution to applied problems Compute probabilities for the Poisson and hypergeometric distributions Find probabilities using a normal distribution table and apply the normal distribution to business problems Recognize

3、when to apply the uniform and exponential distributions,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-3,Probability Distributions,Continuous Probability Distributions,Binomial,Hypergeometric,Poisson,Probability Distributions,Discrete Probability Distributions,No

4、rmal,Uniform,Exponential,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-4,A discrete random variable is a variable that can assume only a countable number of values Many possible outcomes: number of complaints per day number of TVs in a household number of rings

5、before the phone is answered Only two possible outcomes: gender: male or female defective: yes or no spreads peanut butter first vs. spreads jelly first,Discrete Probability Distributions,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-5,Continuous Probability Dis

6、tributions,A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the

7、 ability to measure accurately.,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-6,The Binomial Distribution,Binomial,Hypergeometric,Poisson,Probability Distributions,Discrete Probability Distributions,Business Statistics: A Decision-Making Approach, 6e 2005 Prenti

8、ce-Hall, Inc.,Chap 5-7,The Binomial Distribution,Characteristics of the Binomial Distribution: A trial has only two possible outcomes “success” or “failure” There is a fixed number, n, of identical trials The trials of the experiment are independent of each other The probability of a success, p, rem

9、ains constant from trial to trial If p represents the probability of a success, then (1-p) = q is the probability of a failure,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-8,Binomial Distribution Settings,A manufacturing plant labels items as either defective o

10、r acceptable A firm bidding for a contract will either get the contract or not A marketing research firm receives survey responses of “yes I will buy” or “no I will not” New job applicants either accept the offer or reject it,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, In

11、c.,Chap 5-9,Counting Rule for Combinations,A combination is an outcome of an experiment where x objects are selected from a group of n objects,where: n! =n(n - 1)(n - 2) . . . (2)(1) x! = x(x - 1)(x - 2) . . . (2)(1) 0! = 1 (by definition),Business Statistics: A Decision-Making Approach, 6e 2005 Pre

12、ntice-Hall, Inc.,Chap 5-10,P(x) = probability of x successes in n trials, with probability of success p on each trial x = number of successes in sample, (x = 0, 1, 2, ., n) p = probability of “success” per trial q = probability of “failure” = (1 p) n = number of trials (sample size),P(x),n,x !,n,x,p

13、,q,x,n,x,!,(,),!,=,-,-,Example: Flip a coin four times, let x = # heads: n = 4 p = 0.5 q = (1 - .5) = .5 x = 0, 1, 2, 3, 4,Binomial Distribution Formula,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-11,n = 5 p = 0.1,n = 5 p = 0.5,Mean,0,.2,.4,.6,0,1,2,3,4,5,X,P(

14、X),.2,.4,.6,0,1,2,3,4,5,X,P(X),0,Binomial Distribution,The shape of the binomial distribution depends on the values of p and n,Here, n = 5 and p = .1,Here, n = 5 and p = .5,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-12,Binomial Distribution Characteristics,Me

15、an,Variance and Standard Deviation,Wheren = sample size p = probability of success q = (1 p) = probability of failure,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-13,n = 5 p = 0.1,n = 5 p = 0.5,Mean,0,.2,.4,.6,0,1,2,3,4,5,X,P(X),.2,.4,.6,0,1,2,3,4,5,X,P(X),0,Bi

16、nomial Characteristics,Examples,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-14,Using Binomial Tables,Examples: n = 10, p = .35, x = 3: P(x = 3|n =10, p = .35) = .2522 n = 10, p = .75, x = 2: P(x = 2|n =10, p = .75) = .0004,Business Statistics: A Decision-Makin

17、g Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-15,Using PHStat,Select PHStat / Probability & Prob. Distributions / Binomial,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-16,Using PHStat,Enter desired values in dialog box Here:n = 10 p = .35 Output for x = 0 to x

18、 = 10 will be generated by PHStat Optional check boxes for additional output,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-17,P(x = 3 | n = 10, p = .35) = .2522,PHStat Output,P(x 5 | n = 10, p = .35) = .0949,Business Statistics: A Decision-Making Approach, 6e 20

19、05 Prentice-Hall, Inc.,Chap 5-18,The Poisson Distribution,Binomial,Hypergeometric,Poisson,Probability Distributions,Discrete Probability Distributions,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-19,The Poisson Distribution,Characteristics of the Poisson Distri

20、bution: The outcomes of interest are rare relative to the possible outcomes The average number of outcomes of interest per time or space interval is The number of outcomes of interest are random, and the occurrence of one outcome does not influence the chances of another outcome of interest The prob

21、ability of that an outcome of interest occurs in a given segment is the same for all segments,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-20,Poisson Distribution Formula,where: t = size of the segment of interest x = number of successes in segment of interest

22、= expected number of successes in a segment of unit size e = base of the natural logarithm system (2.71828.),Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-21,Poisson Distribution Characteristics,Mean,Variance and Standard Deviation,where = number of successes in

23、 a segment of unit size t = the size of the segment of interest,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-22,Using Poisson Tables,Example: Find P(x = 2) if = .05 and t = 100,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-

24、23,Graph of Poisson Probabilities,P(x = 2) = .0758,Graphically:, = .05 and t = 100,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-24,Poisson Distribution Shape,The shape of the Poisson Distribution depends on the parameters and t:,t = 0.50,t = 3.0,Business Statis

25、tics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-25,The Hypergeometric Distribution,Binomial,Poisson,Probability Distributions,Discrete Probability Distributions,Hypergeometric,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-26,The Hypergeometr

26、ic Distribution,“n” trials in a sample taken from a finite population of size N Sample taken without replacement Trials are dependent Concerned with finding the probability of “x” successes in the sample where there are “X” successes in the population,Business Statistics: A Decision-Making Approach,

27、 6e 2005 Prentice-Hall, Inc.,Chap 5-27,Hypergeometric Distribution Formula,.,Where N = Population size X = number of successes in the population n = sample size x = number of successes in the sample n x = number of failures in the sample,(Two possible outcomes per trial),Business Statistics: A Decis

28、ion-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-28,Hypergeometric Distribution Formula,Example: 3 Light bulbs were selected from 10. Of the 10 there were 4 defective. What is the probability that 2 of the 3 selected are defective? N = 10n = 3 X = 4 x = 2,Business Statistics: A Decision-Makin

29、g Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-29,Hypergeometric Distribution in PHStat,Select: PHStat / Probability & Prob. Distributions / Hypergeometric ,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-30,Hypergeometric Distribution in PHStat,Complete dialog bo

30、x entries and get output ,N = 10 n = 3 X = 4 x = 2,P(x = 2) = 0.3,(continued),Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-31,The Normal Distribution,Continuous Probability Distributions,Probability Distributions,Normal,Uniform,Exponential,Business Statistics:

31、A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-32,The Normal Distribution,Bell Shaped Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, Spread is determined by the standard deviation, The random variable has an infinite theoretical range: + to ,Mean = Me

32、dian = Mode,x,f(x),Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-33,By varying the parameters and , we obtain different normal distributions,Many Normal Distributions,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-34,The Norm

33、al Distribution Shape,x,f(x),Changing shifts the distribution left or right.,Changing increases or decreases the spread.,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-35,Finding Normal Probabilities,Probability is the area under thecurve!,a,b,x,f(x),P,a,x,b,(,),

34、Probability is measured by the area under the curve,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-36,f(x),x,Probability as Area Under the Curve,0.5,0.5,The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below,Bu

35、siness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-37,Empirical Rules, 1 encloses about 68% of xs,f(x),x,+1,-1,What can we say about the distribution of values around the mean? There are some general rules:,68.26%,Business Statistics: A Decision-Making Approach, 6e 200

36、5 Prentice-Hall, Inc.,Chap 5-38,The Empirical Rule, 2 covers about 95% of xs 3 covers about 99.7% of xs,x,2,2,x,3,3,95.44%,99.72%,(continued),Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-39,Importance of the Rule,If a value is about 2 or more standard deviation

37、s away from the mean in a normal distribution, then it is far from the mean The chance that a value that far or farther away from the mean is highly unlikely, given that particular mean and standard deviation,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-40,The

38、Standard Normal Distribution,Also known as the “z” distribution Mean is defined to be 0 Standard Deviation is 1,z,f(z),0,1,Values above the mean have positive z-values, values below the mean have negative z-values,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-41

39、,The Standard Normal,Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (z) Need to transform x units into z units,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-42,Translation to th

40、e Standard Normal Distribution,Translate from x to the standard normal (the “z” distribution) by subtracting the mean of x and dividing by its standard deviation:,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-43,Example,If x is distributed normally with mean of

41、100 and standard deviation of 50, the z value for x = 250 is This says that x = 250 is three standard deviations (3 increments of 50 units) above the mean of 100.,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-44,Comparing x and z units,z,100,3.0,0,250,x,Note tha

42、t the distribution is the same, only the scale has changed. We can express the problem in original units (x) or in standardized units (z), = 100 = 50,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-45,The Standard Normal Table,The Standard Normal table in the text

43、book (Appendix D) gives the probability from the mean (zero) up to a desired value for z,z,0,2.00,.4772,Example: P(0 z 2.00) = .4772,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-46,The Standard Normal Table,The value within the table gives the probability from

44、z = 0 up to the desired z value,.4772,2.0,P(0 z 2.00) = .4772,The row shows the value of z to the first decimal point,The column gives the value of z to the second decimal point,2.0,. . .,(continued),Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-47,General Proce

45、dure for Finding Probabilities,Draw the normal curve for the problem in terms of x Translate x-values to z-values Use the Standard Normal Table,To find P(a x b) when x is distributed normally:,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-48,Z Table example,Supp

46、ose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 x 8.6),P(8 x 8.6) = P(0 z 0.12),Z,0.12,0,x,8.6,8,Calculate z-values:,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-49,Z Table example,Suppose x is normal with mean 8.0 and standard deviation 5.0.

47、 Find P(8 x 8.6),P(0 z 0.12),z,0.12,0,x,8.6,8,P(8 x 8.6),= 8 = 5,= 0 = 1,(continued),Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-50,Z,0.12,z,.00,.01,0.0,.0000,.0040,.0080,.0398,.0438,0.2,.0793,.0832,.0871,0.3,.1179,.1217,.1255,Solution: Finding P(0 z 0.12),.04

48、78,.02,0.1,.0478,Standard Normal Probability Table (Portion),0.00,= P(0 z 0.12),P(8 x 8.6),Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-51,Finding Normal Probabilities,Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x 8.6),Z,8.6,8.0,Bus

49、iness Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-52,Finding Normal Probabilities,Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x 8.6),(continued),Z,0.12,.0478,0.00,.5000,P(x 8.6) = P(z 0.12) = P(z 0) + P(0 z 0.12) = .5 + .0478 = .5478,Busine

50、ss Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-53,Upper Tail Probabilities,Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x 8.6),Z,8.6,8.0,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-54,Now Find P(x 8.6)

51、,(continued),Z,0.12,0,Z,0.12,.0478,0,.5000,.50 - .0478 = .4522,P(x 8.6) = P(z 0.12) = P(z 0) - P(0 z 0.12) = .5 - .0478 = .4522,Upper Tail Probabilities,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-55,Lower Tail Probabilities,Suppose x is normal with mean 8.0 a

52、nd standard deviation 5.0. Now Find P(7.4 x 8),Z,7.4,8.0,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-56,Lower Tail Probabilities,Now Find P(7.4 x 8),Z,7.4,8.0,The Normal distribution is symmetric, so we use the same table even if z-values are negative: P(7.4 x

53、 8) = P(-0.12 z 0) = .0478,(continued),.0478,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-57,Normal Probabilities in PHStat,We can use Excel and PHStat to quickly generate probabilities for any normal distribution We will find P(8 x 8.6) when x is normally dist

54、ributed with mean 8 and standard deviation 5,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-58,PHStat Dialogue Box,Select desired options and enter values,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-59,PHStat Output,Busines

55、s Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-60,The Uniform Distribution,Continuous Probability Distributions,Probability Distributions,Normal,Uniform,Exponential,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-61,The Uniform Distri

56、bution,The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-62,The Continuous Uniform Distribution:,where f(x) = value of the density fu

57、nction at any x value a = lower limit of the interval b = upper limit of the interval,The Uniform Distribution,(continued),f(x) =,Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-63,Uniform Distribution,Example: Uniform Probability Distribution Over the range 2 x 6:,2,6,.25,f(x) = = .25 for 2 x 6,6 - 2,1,x,f(x),Business Statistics: A Decision-Making Approach, 6e 2005 Prentice-Hall, Inc.,Chap 5-64,The Exponential Distribution,Continuous Probabi