Chapter 6 Continuous Probability Distributions Business Statistics: A Decision-Making Approach 6th Edition Chapter 6 Continuous Probability Distributions Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Probability Distributions The Normal Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Exponential Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

The Normal Distribution ‘Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   f(x) σ x μ Mean = Median = Mode Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Many Normal Distributions By varying the parameters μ and σ, we obtain different normal distributions Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

The Normal Distribution Shape f(x) Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread. σ μ x Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Finding Normal Probabilities Probability is the area under the curve! Probability is measured by the area under the curve f(x) P ( a  x  b ) a b x Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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

Empirical Rules What can we say about the distribution of values around the mean? There are some general rules: f(x) μ ± 1σ encloses about 68% of x’s σ σ x μ-1σ μ μ+1σ 68.26% Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

The Empirical Rule μ ± 2σ covers about 95% of x’s (continued) μ ± 2σ covers about 95% of x’s μ ± 3σ covers about 99.7% of x’s 3σ 3σ 2σ 2σ μ x μ x 95.44% 99.72% Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Importance of the Rule If a value is about 2 or more standard deviations 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.

The Standard Normal Distribution Also known as the “z” distribution Mean is defined to be 0 Standard Deviation is 1 f(z) 1 z 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.

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.

Translation to the 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.

Example If x is distributed normally with mean of 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.

Comparing x and z units 100 250 x 3.0 z μ = 100 σ = 50 3.0 z Note that the distribution is the same, only the scale has changed. We can express the problem in original units (x) or in standardized units (z) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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

The Standard Normal Table (continued) The column gives the value of z to the second decimal point The row shows the value of z to the first decimal point The value within the table gives the probability from z = 0 up to the desired z value . 2.0 .4772 P(0 < z < 2.00) = .4772 2.0 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

General Procedure for Finding Probabilities To find P(a < x < b) when x is distributed normally: Draw the normal curve for the problem in terms of x Translate x-values to z-values Use the Standard Normal Table Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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

Z Table example (continued) Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) = 8  = 5 = 0  = 1 x z 8 8.6 0.12 P(8 < x < 8.6) P(0 < z < 0.12) Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Solution: Finding P(0 < z < 0.12) Standard Normal Probability Table (Portion) P(8 < x < 8.6) = P(0 < z < 0.12) .02 z .00 .01 .0478 0.0 .0000 .0040 .0080 0.1 .0398 .0438 .0478 0.2 .0793 .0832 .0871 Z 0.3 .1179 .1217 .1255 0.00 0.12 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Finding Normal Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) Z 8.0 8.6 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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

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

Upper Tail Probabilities (continued) Now Find P(x > 8.6)… P(x > 8.6) = P(z > 0.12) = P(z > 0) - P(0 < z < 0.12) = .5 - .0478 = .4522 .0478 .5000 .50 - .0478 = .4522 Z Z 0.12 0.12 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Lower Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < x < 8) Z 8.0 7.4 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Lower Tail Probabilities (continued) Now Find P(7.4 < x < 8)… The Normal distribution is symmetric, so we use the same table even if z-values are negative: P(7.4 < x < 8) = P(-0.12 < z < 0) = .0478 .0478 Z 8.0 7.4 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

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 distributed with mean 8 and standard deviation 5 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Select desired options and enter values PHStat Dialogue Box Select desired options and enter values Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

PHStat Output Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Probability Distributions The Uniform Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Exponential Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

The Uniform Distribution 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.

The Uniform Distribution (continued) The Continuous Uniform Distribution: f(x) = where f(x) = value of the density function at any x value a = lower limit of the interval b = upper limit of the interval Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Uniform Distribution Example: Uniform Probability Distribution Over the range 2 ≤ x ≤ 6: 1 f(x) = = .25 for 2 ≤ x ≤ 6 6 - 2 f(x) .25 x 2 6 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Probability Distributions The Exponential Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Exponential Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

The Exponential Distribution Used to measure the time that elapses between two occurrences of an event (the time between arrivals) Examples: Time between trucks arriving at an unloading dock Time between transactions at an ATM Machine Time between phone calls to the main operator Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

The Exponential Distribution The probability that an arrival time is equal to or less than some specified time a is where 1/ is the mean time between events Note that if the number of occurrences per time period is Poisson with mean , then the time between occurrences is exponential with mean time 1/  Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Exponential Distribution (continued) Shape of the exponential distribution f(x)  = 3.0 (mean = .333)  = 1.0 (mean = 1.0) = 0.5 (mean = 2.0) x Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Example Example: Customers arrive at the claims counter at the rate of 15 per hour (Poisson distributed). What is the probability that the arrival time between consecutive customers is less than five minutes? Time between arrivals is exponentially distributed with mean time between arrivals of 4 minutes (15 per 60 minutes, on average) 1/ = 4.0, so  = .25 P(x < 5) = 1 - e-a = 1 – e-(.25)(5) = .7135 Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.

Chapter Summary Reviewed key discrete distributions binomial, poisson, hypergeometric Reviewed key continuous distributions normal, uniform, exponential Found probabilities using formulas and tables Recognized when to apply different distributions Applied distributions to decision problems Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc.