11/16/2017. Chapter. Copyright 2009 by The McGraw-Hill Companies, Inc. 7-2

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1 7 Chapter Continuous Probability Distributions Describing a Continuous Distribution Uniform Continuous Distribution Normal Distribution Normal Approximation to the Binomial Normal Approximation to the Poisson Exponential Distribution Triangular Distribution McGraw-Hill/Irwin Copyright 2009 by The McGraw-Hill Companies, Inc. Continuous Variables Events as Intervals Discrete Variable each value of X has its own probability P(X). Continuous Variable events are intervals and probabilities are areas underneath smooth curves. A single point has no probability

2 Describing a Continuous Distribution PDFs and CDFs Probability Density Function (PDF) For a continuous random variable, the PDF is an equation that shows the height of the curve f(x) at each possible value of X over the range of X. 7-3 Describing a Continuous Distribution PDFs and CDFs Continuous PDF s: Denoted f(x) Must be nonnegative Total area under curve = 1 Mean, variance and shape depend on the PDF parameters Reveals the shape of the distribution Normal PDF 7-4 2

3 Describing a Continuous Distribution PDFs and CDFs Continuous CDF s: Denoted F(x) Shows P(X < x), the cumulative proportion of scores Useful for finding probabilities 7-5 Describing a Continuous Distribution Probabilities as Areas Continuous probability functions are smooth curves. Unlike discrete distributions, the area at any single point = 0. The entire area under any PDF must be 1. Mean is the balance point of the distribution

4 Describing a Continuous Distribution Expected Value and Variance 7-7 Uniform Continuous Distribution Characteristics of the Uniform Distribution If X is a random variable that is uniformly distributed between a and b, its PDF has constant height. Denoted U(a,b) Area = base x height = (b-a) x 1/(b-a) =

5 Uniform Continuous Distribution Characteristics of the Uniform Distribution 7-9 Uniform Continuous Distribution Characteristics of the Uniform Distribution The CDF increases linearly to 1. CDF formula is (x-a)/( )/(b b-a)

6 Uniform Continuous Distribution Example: Anesthesia Effectiveness An oral surgeon injects a painkiller prior to extracting a tooth. Given the varying characteristics of patients, the dentist views the time for anesthesia effectiveness as a uniform random variable that takes between 15 minutes and 30 minutes. X is U(15 15,, 30 30)) a = 15 15,, b = 30 30,, find the mean and standard deviation Uniform Continuous Distribution Example: Anesthesia Effectiveness m= a + b = = minutes ))2 = 4.33 minutes (b a)2 = (30 15 s= Find the probability that the anesthetic takes between 20 and 25 minutes. P(c < X < d) = (d (d c)/( )/(b b a) P(20 < X < 25 25)) = ( )/( )/( )) = 5/15 = or % %

7 Uniform Continuous Distribution Example: Anesthesia Effectiveness P(20 < X < 25 25)) 7-13 Normal Distribution Characteristics of the Normal Distribution Normal or Gaussian distribution was named for German mathematician Karl Gauss (1777 ( ). ). Defined by two parameters, mand s Denoted N(m, s) Domain is < X < + Almost all area under the normal curve is included in the range m 3s< X < m+ 3s

8 Normal Distribution Characteristics of the Normal Distribution 7-15 Normal Distribution Characteristics of the Normal Distribution Normal PDF f(x) reaches a maximum at m and has points of inflection at m+ s Bell-shaped curve Figure

9 Normal Distribution Characteristics of the Normal Distribution Normal CDF Figure Normal Distribution Characteristics of the Normal Distribution All normal distributions have the same shape but differ in the axis scales. µ = 42.70mm σ = 0.01mm Diameters of golf balls µ = 70 σ = 10 CPA Exam Scores

10 Normal Distribution What is Normal? A normal random variable should: Be measured on a continuous scale. Possess clear central tendency. Have only one peak (unimodal (unimodal). ). Exhibit tapering tails. Be symmetric about the mean (equal tails) Characteristics of the Standard Normal Since for every value of mand s, there is a different normal distribution, we transform a normal random variable to a standard normal distribution with m= 0 and s= 1 using the formula: z= x m s Denoted N(0,1)

11 Characteristics of the Standard Normal 7-21 Characteristics of the Standard Normal Standard normal PDF f(x) reaches a maximum at 0 and has points of inflection at +1. Shape is unaffected by the transformation. It is still a bellbellshaped curve. Figure

12 Characteristics of the Standard Normal Standard normal CDF Figure Characteristics of the Standard Normal A common scale from -3 to is used. Entire area under the curve is unity. The probability of an event P(z1 < Z < z2) is a definite integral of f(z). However, standard normal tables or Excel functions can be used to find the desired probabilities

13 Normal Areas from Appendix CC-1 Appendix CC-1 allows you to find the area under the curve from 0 to z. For example, find P(0 < Z < ): ): Figure Table

14 Normal Areas from Appendix CC-1 Now find P(Z < ): ): = Normal Areas from Appendix CC-1 Now find P(-1.96 < Z < ). ). Due to symmetry, P(-1.96 < Z) is the same as P(Z < ). ) So, P(-1.96 < Z < )) = =.9500 or 95 95% % of the area under the curve

15 Basis for the Empirical Rule Approximately 95 95% % of the area under the curve is between + 2s Approximately % of the area under the curve is between + 3s 68.26% Figure Normal Areas from Appendix CC-2 Appendix CC-2 allows you to find the area under the curve from the left of z (similar to Excel). For example,.9500 P(Z < 1.96) P(Z < )) P(-1.96 < Z < )) Figure

16 Table Normal Areas from Appendices CC-1 or CC-2 Appendices CC-1 and CC-2 yield identical results. Use whichever table is easiest. Finding z for a Given Area Appendices CC-1 and CC-2 be used to find the z-value corresponding to a given probability. For example, what z-value defines the top 1% of a normal distribution? This implies that 49 49% % of the area lies between 0 and z

17 Finding z for a Given Area Look for an area of.4900 in Appendix C-1: Without interpolation, the closest we can get is z = Finding z for a Given Area Some important Normal areas: Table

18 Finding Normal Areas with Excel 7-35 Finding Normal Areas with Excel

19 Finding Normal Areas with Excel 7-37 Finding Normal Areas with Excel

20 Finding Areas by using Standardized Variables Suppose John took an economics exam and scored 86 points. The class mean was 75 with a standard deviation of 7. What percentile is John in (i.e., find P(X < 86 86)? )? zjohn = x m = = 11/7 = s So John s score is 1.57 standard deviations about the mean Finding Areas by using Standardized Variables P(X < 86 86)) = P(Z < )) = (from Appendix CC-2) So, John is approximately in the 94th percentile Figure

21 Inverse Normal You can manipulate the transformation formula to find the normal percentile values (e.g., 5th, 10th, 25th, etc.): x = m+ zs Here are some common percentiles 7-41 Using Excel Without Standardizing Excel s NORMDIST and NORMINV function allow you to evaluate areas without standardizing. For example, let m= cm and s= cm, what is the probability that a given steel bearing will have a diameter between and cm? cm? In other words, P(2.039 < X < )) Excel only gives left tail areas, so break the formula into two, find P(X < )) and P(X < ), ), then subtract them to find the desired probability:

22 Using Excel Without Standardizing P(X < )) = P(X < )) = P(2.039 < X < )) = = or % 7-43 Normal Approximation to the Binomial When is Approximation Needed? Binomial probabilities are difficult to calculate when n is large. Use a normal approximation to the binomial. As n becomes large, the binomial bars become more continuous and smooth

23 Normal Approximation to the Binomial When is Approximation Needed? Rule of thumb: when np> 10 and n(1-p) > 10,, then it is appropriate to use the normal 10 approximation to the binomial. In this case, the binomial mean and standard deviation will be equal to the normal mand s, respectively. m= np s= np(1-p) 7-45 Normal Approximation to the Binomial Example Coin Flips If we were to flip a coin n = 32 times and p=.50.50,, are the requirements for a normal approximation to the binomial met? Are np> 10 and n(1-p) > 10 10?? np= 32 x = 16 n(1-p) = 32 x (1 ( )) = 16 So, a normal approximation can be used. When translating a discrete scale into a continuous scale, care must be taken about individual points

24 Normal Approximation to the Binomial Example Coin Flips For example, find the probability of more than 17 heads in 32 flips of a fair coin. This can be written as P(X > 18 18). ). However, more than actually falls between 17 and 18 on a discrete scale. Figure Normal Approximation to the Binomial Example Coin Flips Since the cutoff point for more than is halfway between 17 and 18 18,, we add 0.5 to the lower limit and find P(X > ). This addition to X is called the Continuity Correction.. Correction At this point, the problem can be completed as any normal distribution problem

25 Normal Approximation to the Binomial Continuity Correction The table below shows some events and their cutoff point for the normal approximation Normal Approximation to the Poisson When is Approximation Needed? The normal approximation to the Poisson works best when lis large (e.g., when l exceeds the values in Appendix B). Set the normal mand sequal to the Poisson mean and standard deviation. m= l s= l

26 Normal Approximation to the Poisson Example Utility Bills On Wednesday between 10A.M. and noon customer billing inquiries arrive at a mean rate of 42 inquiries per hour at Consumers Energy. What is the probability of receiving more than 50 calls? l= 42 which is too big to use the Poisson table. Use the normal approximation with m= l= 42 s= l = 42 = Normal Approximation to the Poisson Example Utility Bills To find P(X > 50 50)) calls, use the continuitycontinuitycorrected cutoff point halfway between 50 and 51 (i.e., X = ). At this point, the problem can be completed as any normal distribution problem

27 Exponential Distribution Characteristics of the Exponential Distribution If events per unit of time follow a Poisson distribution, the waiting time until the next event follows the Exponential distribution. Waiting time until the next event is a continuous variable Exponential Distribution Characteristics of the Exponential Distribution

28 Exponential Distribution Characteristics of the Exponential Distribution Probability of waiting more than x Probability of waiting less than x 7-55 Exponential Distribution Example Customer Waiting Time Between 2P.M. and 4P.M. on Wednesday, patient insurance inquiries arrive at Blue Choice insurance at a mean rate of 2.2 calls per minute. What is the probability of waiting more than 30 seconds (i.e., 0.50 minutes) for the next call? Set l= 2.2 events/min and x = 0.50 min P(X > )) = e lx = e (2.2)()(00.5) = or % % chance of waiting more than 30 seconds for the next call

29 Exponential Distribution Example Customer Waiting Time P(X > 0.50) P(X < 0.50) 7-57 Exponential Distribution Inverse Exponential If the mean arrival rate is 2.2 calls per minute, we want the 90th percentile for waiting time (the top 10 10% % of waiting time). Find the x-value that defines the upper 10 10%. %

30 Exponential Distribution Inverse Exponential P(X < x) = or P(X > x) = So, e lx = lx = ln ln(. (.10 10)) = x = //l = //2.2 = min. 90% 90 % of the calls will arrive within minutes (62 (62..8 seconds) Exponential Distribution Inverse Exponential Quartiles for Exponential with l= 2.2 Table

31 Exponential Distribution Mean Time Between Events Exponential waiting times are described as Mean time between events (MTBE) = 1/l 1/MTBE = l= mean events per unit of time In a hospital, if an event is patient arrivals in an ER, and the MTBE is 20 minutes, then l= 1/20 = 0.05 arrivals per minute (or 3/hour) Exponential Distribution Using Excel In Excel, use =EXPONDIST(x =EXPONDIST(x,l,1) to return the leftleft-tail area P(X < x). Relation Between Exponential and Poisson Table

32 Triangular Distribution Characteristics of the Triangular Distribution A simple distribution that can be symmetric or skewed. Ranges from a to b and has a mode or peak at c Denoted T(a,b,c) 7-63 Triangular Distribution Characteristics of the Triangular Distribution Table

33 Triangular Distribution Special Cases: Symmetric Triangular You can easily generate random triangular data T(0, 1, 2) in Excel by summing RAND()+RAND(). The triangular distribution T( ( ,, 0, )) closely resembles a standard normal distribution N(0 N(0, 1) Continuous Distributions Compared

34 Applied Statistics in Business & Economics End of Chapter