Prepared by Lloyd R. Jaisingh A PowerPoint Presentation Package to Accompany Applied Statistics in Business & Economics, 4th edition David P. Doane and Lori E. Seward Prepared by Lloyd R. Jaisingh

Continuous Probability Distributions Chapter 7 Chapter Contents 7.1 Describing a Continuous Distribution 7.2 Uniform Continuous Distribution 7.3 Normal Distribution 7.4 Standard Normal Distribution 7.5 Normal Approximations 7.6 Exponential Distribution 7.7 Triangular Distribution (Optional)

Continuous Probability Distributions Chapter 7 Chapter Learning Objectives LO7-1: Define a continuous random variable. LO7-2: Calculate uniform probabilities. LO7-3: Know the form and parameters of the normal distribution. LO7-4: Find the normal probability for given x using tables or Excel. LO7-5: Solve for z or x for a given normal probability using tables or Excel. LO7-6: Use the normal approximation to a binomial or a Poisson. LO7-7: Find the exponential probability for a given x. LO7-8: Solve for x for given exponential probability. LO7-9: Use the triangular distribution for “what-if” analysis (optional).

Events as Intervals 7.1 Describing a Continuous Distribution Chapter 7 LO7-1 Chapter 7 LO7-1: Define a continuous random variable. 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.

PDF – Probability Density Function 7.1 Describing a Continuous Distribution LO7-1 Chapter 7 PDF – Probability Density Function 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 CDF – Cumulative Distribution Function Continuous CDF’s: Denoted F(x). Shows P(X ≤ x), the cumulative proportion of scores. Useful for finding probabilities. Normal CDF

Probabilities as Areas 7.1 Describing a Continuous Distribution LO7-1 Chapter 7 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. LO7-2: Calculate uniform probabilities. 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) = 1.

7.2 Uniform Continuous Distribution LO7-2 Chapter 7 LO7-2: Calculate uniform probabilities. Characteristics of the Uniform Distribution

Characteristics of the Normal Distribution LO7-3 7.3 Normal Distribution Chapter 7 LO7-3: Know the form and parameters of the normal distribution. Characteristics of the Normal Distribution 1. Normal or Gaussian (or bell shaped) distribution was named for German mathematician Karl Gauss (1777 – 1855). 2. Defined by two parameters, µ and . Denoted by N(µ, ). 3. Domain is –  < X < +  (continuous scale) 4. Almost all area under the normal curve is included in the range µ – 3 < X < µ + 3. 5. Symmetric and unimodal about the mean.

Characteristics of the Standard Normal Distribution LO7-3 7.4 Standard Normal Distribution Chapter 7 Characteristics of the Standard Normal Distribution Since for every value of µ and , there is a different normal distribution, we transform a normal random variable to a standard normal distribution with µ = 0 and  = 1 using the formula.

Normal Areas from Appendices C-1 or C-2 LO7-4 7.4 Standard Normal Distribution Chapter 7 LO7-4: Find the normal probability for given z or x using tables or Excel. Normal Areas from Appendices C-1 or C-2 Appendices C-1 and C-2 will yield identical results. Use whichever table is easiest. Finding z for a Given Area Appendices C-1 and C-2 be used to find the z-value corresponding to a given probability. LO7-5: Solve for z or x for a normal probability using tables or Excel. Inverse Normal How can we find the various normal percentiles (5th, 10th, 25th, 75th, 90th, 95th, etc.) known as the inverse normal? That is, how can we find X for a given area? We simply turn the standardizing transformation around: x = μ + zσ solving for x in z = (x − μ)/. Figure 7.15

7.4 Standard Normal Distribution LO7-5 7.4 Standard Normal Distribution Chapter 7 Inverse Normal Figure 7.18

When is Approximation Needed? LO7-6 7.5 Normal Approximations Chapter 7 LO7-6: Use the normal approximation to a binomial or a Poisson. When is Approximation Needed? Rule of thumb: when n > 10 and n(1- ) > 10, then it is appropriate to use the normal approximation to the binomial. In this case, the binomial mean and standard deviation will be equal to the normal µ and , respectively. When is Approximation Needed? The normal approximation to the Poisson works best when  is large (e.g., when  exceeds the values in Appendix B). Set the normal µ and  equal to the Poisson mean and standard deviation.

Characteristics of the Exponential Distribution LO7-7 7.6 Exponential Distribution Chapter 7 LO7-7: Find the exponential probability for a given x. 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.

Characteristics of the Exponential Distribution LO7-7 7.6 Exponential Distribution Chapter 7 Characteristics of the Exponential Distribution Probability of waiting more than x Probability of waiting less than x LO7-8: Solve for x for given exponential probability. Inverse Exponential For example, if the mean arrival rate is 2.2 calls per minute, we want the 90th percentile for waiting time (the top 10% of waiting time). Find the x-value that defines the upper 10%.

Characteristics of the Triangular Distribution LO7-9 7.7 Triangular Distribution Chapter 7 LO7-9: Use the triangular distribution for “what-if” analysis (optional). Characteristics of the Triangular Distribution

Characteristics of the Triangular Distribution LO7-9 7.7 Triangular Distribution Chapter 7 Characteristics of the Triangular Distribution The triangular distribution is a way of thinking about variation that corresponds rather well to what-if analysis in business. It is not surprising that business analysts are attracted to the triangular model. Its finite range and simple form are more understandable than a normal distribution. It is more versatile than a normal, because it can be skewed in either direction. Yet it has some of the nice properties of a normal, such as a distinct mode. The triangular model is especially handy for what-if analysis when the business case depends on predicting a stochastic variable (e.g., the price of a raw material, an interest rate, a sales volume). If the analyst can anticipate the range (a to c) and most likely value (b), it will be possible to calculate probabilities of various outcomes. Many times, such distributions will be skewed, so a normal wouldn’t be much help.