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Continuous Probability Distributions f (x) Uniform Probability Distribution Normal Probability Distribution Exponential Probability Distribution f (x) Exponential Uniform f (x) Normal x x x Continuous Probability Distributions A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. It is not possible to talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the random variable assuming a value within a given interval. Continuous Probability Distributions f (x) The probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2. f (x) Exponential Uniform f (x) x1 x 2 Normal x1 xx12 x2 x x1 x 2 x x Uniform Probability Distribution A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. The uniform probability density function is: f (x) = 1/(b – a) for a < x < b =0 elsewhere where: a = smallest value the variable can assume b = largest value the variable can assume Uniform Probability Distribution Expected Value of x E(x) = (a + b)/2 Variance of x Var(x) = (b - a)2/12 Example: Slater's Buffet Uniform Probability Distribution Slater customers are charged for the amount of salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces. Example: Slater's Buffet Uniform Probability Density Function f(x) = 1/10 for 5 < x < 15 =0 elsewhere where: x = salad plate filling weight Example: Slater's Buffet Expected Value of x E(x) = (a + b)/2 = (5 + 15)/2 = 10 Variance of x Var(x) = (b - a)2/12 = (15 – 5)2/12 = 8.33 Example: Slater's Buffet Uniform Probability Distribution for Salad Plate Filling Weight f(x) 1/10 5 10 15 Salad Weight (oz.) x Example: Slater's Buffet What is the probability that a customer will take between 12 and 15 ounces of salad? f(x) P(12 < x < 15) = 1/10(3) = .3 1/10 5 10 12 15 Salad Weight (oz.) x Normal Probability Distribution The normal probability distribution is the most important distribution for describing a continuous random variable. It is widely used in statistical inference. Normal Probability Distribution It has been used in a wide variety of applications: Heights of people Scientific measurements Normal Probability Distribution It has been used in a wide variety of applications: Test scores Amounts of rainfall Normal Probability Distribution Normal Probability Density Function 1 ( x )2 /2 2 f (x) e 2 where: = mean = standard deviation = 3.14159 e = 2.71828 Normal Probability Distribution Characteristics The distribution is symmetric, and is bell-shaped. x Normal Probability Distribution Characteristics The entire family of normal probability distributions is defined by its mean and its standard deviation . Standard Deviation Mean x Normal Probability Distribution Characteristics The highest point on the normal curve is at the mean, which is also the median and mode. x Normal Probability Distribution Characteristics The mean can be any numerical value: negative, zero, or positive. x -10 0 20 Normal Probability Distribution Characteristics The standard deviation determines the width of the curve: larger values result in wider, flatter curves. = 15 = 25 x Normal Probability Distribution Characteristics Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and .5 to the right). .5 .5 x Normal Probability Distribution Characteristics 68.26% of values of a normal random variable are within +/- 1 standard deviation of its mean. 95.44% of values of a normal random variable are within +/- 2 standard deviations of its mean. 99.72% of values of a normal random variable are within +/- 3 standard deviations of its mean. Normal Probability Distribution Characteristics 99.72% 95.44% 68.26% – 3 – 1 – 2 + 3 + 1 + 2 x Standard Normal Probability Distribution A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability distribution. Standard Normal Probability Distribution The letter z is used to designate the standard normal random variable. 1 z 0 Standard Normal Probability Distribution Converting to the Standard Normal Distribution z x We can think of z as a measure of the number of standard deviations x is from . Using Excel to Compute Standard Normal Probabilities Excel has two functions for computing probabilities and z values for a standard normal distribution: NORM S DIST is used to compute the cumulative NORMSDIST probability given a z value. NORMSINV NORM S INV is used to compute the z value given a cumulative probability. (The “S” in the function names reminds us that they relate to the standard normal probability distribution.) Using Excel to Compute Standard Normal Probabilities Formula Worksheet 1 2 3 4 5 6 7 8 9 A B Probabilities: Standard Normal Distribution P (z < 1.00) P (0.00 < z < 1.00) P (0.00 < z < 1.25) P (-1.00 < z < 1.00) P (z > 1.58) P (z < -0.50) =NORMSDIST(1) =NORMSDIST(1)-NORMSDIST(0) =NORMSDIST(1.25)-NORMSDIST(0) =NORMSDIST(1)-NORMSDIST(-1) =1-NORMSDIST(1.58) =NORMSDIST(-0.5) Using Excel to Compute Standard Normal Probabilities Value Worksheet 1 2 3 4 5 6 7 8 9 A B Probabilities: Standard Normal Distribution P (z < 1.00) P (0.00 < z < 1.00) P (0.00 < z < 1.25) P (-1.00 < z < 1.00) P (z > 1.58) P (z < -0.50) 0.8413 0.3413 0.3944 0.6827 0.0571 0.3085 Using Excel to Compute Standard Normal Probabilities Formula Worksheet 1 2 3 4 5 6 A B Finding z Values, Given Probabilities z value with .10 in upper tail z value with .025 in upper tail z value with .025 in lower tail =NORMSINV(0.9) =NORMSINV(0.975) =NORMSINV(0.025) Using Excel to Compute Standard Normal Probabilities Value Worksheet 1 2 3 4 5 6 A B Finding z Values, Given Probabilities z value with .10 in upper tail z value with .025 in upper tail z value with .025 in lower tail 1.28 1.96 -1.96 Example: Pep Zone Standard Normal Probability Distribution Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to Pep Zone 20 gallons, a replenishment order is 5w-20 placed. Motor Oil Example: Pep Zone Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution The store manager is concerned that sales are being lost due to stockouts while waiting for an order. It has been determined that demand during replenishment leadtime is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. The manager would like to know the probability of a stockout, P(x > 20). Example: Pep Zone Pep Zone Solving for the Stockout Probability 5w-20 Motor Oil Step 1: Convert x to the standard normal distribution. z = (x - )/ = (20 - 15)/6 = .83 Step 2: Find the area under the standard normal curve between the mean and z = .83. see next slide Example: Pep Zone Pep Zone 5w-20 Motor Oil Probability Table for the Standard Normal Distribution z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 . . . . . . . . . . . .5 .1915 .1695 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 .6 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2517 .2549 .7 .2580 .2611 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852 .8 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 .9 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389 . . . . . . . P(0 < z < .83) . . . . Example: Pep Zone Pep Zone Solving for the Stockout Probability 5w-20 Motor Oil Step 3: Compute the area under the standard normal curve to the right of z = .83. P(z > .83) = .5 – P(0 < z < .83) = 1- .2967 = .2033 Probability of a stockout P(x > 20) Example: Pep Zone Pep Zone 5w-20 Motor Oil Solving for the Stockout Probability Area = .5 - .2967 Area = .2967 = .2033 0 .83 z Example: Pep Zone Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution If the manager of Pep Zone wants the probability of a stockout to be no more than .05, what should the reorder point be? Example: Pep Zone Pep Zone 5w-20 Motor Oil Solving for the Reorder Point Area = .4500 Area = .0500 0 z.05 z Example: Pep Zone Pep Zone 5w-20 Motor Oil Solving for the Reorder Point Step 1: Find the z-value that cuts off an area of .05 in the right tail of the standard normal distribution. z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 . . . . . . . . . . . 1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441 1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545 1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633 1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706 1.9 .4713 .4719 .4726 We .4732look .4738 .4750 .4756 .4761 .4767 up.4744 the area . . . . . (.5 - ..05 = .45) . . . . . Example: Pep Zone Pep Zone Solving for the Reorder Point 5w-20 Motor Oil Step 2: Convert z.05 to the corresponding value of x. x = + z.05 = 15 + 1.645(6) = 24.87 or 25 A reorder point of 25 gallons will place the probability of a stockout during leadtime at (slightly less than) .05. Example: Pep Zone Pep Zone 5w-20 Motor Oil Solving for the Reorder Point By raising the reorder point from 20 gallons to 25 gallons on hand, the probability of a stockout decreases from about .20 to .05. This is a significant decrease in the chance that Pep Zone will be out of stock and unable to meet a customer’s desire to make a purchase. Using Excel to Compute Normal Probabilities Excel has two functions for computing cumulative probabilities and x values for any normal distribution: NORMDIST is used to compute the cumulative probability given an x value. NORMINV is used to compute the x value given a cumulative probability. Using Excel to Compute Normal Probabilities Formula Worksheet 1 2 3 4 5 6 7 8 Pep Zone 5w-20 Motor Oil A B Probabilities: Normal Distribution P (x > 20) =1-NORMDIST(20,15,6,TRUE) Finding x Values, Given Probabilities x value with .05 in upper tail =NORMINV(0.95,15,6) Using Excel to Compute Normal Probabilities 5w-20 Motor Oil Value Worksheet 1 2 3 4 5 6 7 8 Pep Zone A B Probabilities: Normal Distribution P (x > 20) 0.2023 Finding x Values, Given Probabilities x value with .05 in upper tail 24.87 Note: P(x > 20) = .2023 here using Excel, while our previous manual approach using the z table yielded .2033 due to our rounding of the z value. Exponential Probability Distribution The exponential probability distribution is useful in describing the time it takes to complete a task. The exponential random variables can be used to describe: Time between vehicle arrivals at a toll booth Time required to complete a questionnaire Distance between major defects in a highway Exponential Probability Distribution Density Function f ( x) 1 e x / for x > 0, > 0 where: = mean e = 2.71828 Exponential Probability Distribution Cumulative Probabilities P ( x x0 ) 1 e xo / where: x0 = some specific value of x Using Excel to Compute Exponential Probabilities The EXPONDIST function can be used to compute exponential probabilities. The EXPONDIST function has three arguments: 1st The value of the random variable x 2nd 1/m 3rd the inverse of the mean number of occurrences “TRUE” or “FALSE” in an interval we will always enter “TRUE” because we’re seeking a cumulative probability Using Excel to Compute Exponential Probabilities Formula Worksheet A 1 2 3 P (x < 18) 4 P (6 < x < 18) 5 P (x > 8) 6 B Probabilities: Exponential Distribution =EXPONDIST(18,1/15,TRUE) =EXPONDIST(18,1/15,TRUE)-EXPONDIST(6,1/15,TRUE) =1-EXPONDIST(8,1/15,TRUE) Using Excel to Compute Exponential Probabilities Value Worksheet A 1 2 3 P (x < 18) 4 P (6 < x < 18) 5 P (x > 8) 6 B Probabilities: Exponential Distribution 0.6988 0.3691 0.5866 Example: Al’s Full-Service Pump Exponential Probability Distribution The time between arrivals of cars at Al’s full-service gas pump follows an exponential probability distribution with a mean time between arrivals of 3 minutes. Al would like to know the probability that the time between two successive arrivals will be 2 minutes or less. Example: Al’s Full-Service Pump Exponential Probability Distribution f(x) P(x < 2) = 1 - 2.71828-2/3 = 1 - .5134 = .4866 .4 .3 .2 .1 x 1 2 3 4 5 6 7 8 9 10 Time Between Successive Arrivals (mins.) Using Excel to Compute Exponential Probabilities Formula Worksheet 1 2 3 4 A B Probabilities: Exponential Distribution P (x < 2) =EXPONDIST(2,1/3,TRUE) Using Excel to Compute Exponential Probabilities Value Worksheet 1 2 3 4 A B Probabilities: Exponential Distribution P (x < 2) 0.4866 Relationship between the Poisson and Exponential Distributions The Poisson distribution provides an appropriate description of the number of occurrences per interval The exponential distribution provides an appropriate description of the length of the interval between occurrences