Continuous Probability Distributions

Chapter 06 Continuous Probability Distributions Part 06 6.1 Uniform Probability Distribution 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 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) Uniform x1 x2 f (x) x f (x) Exponential Normal x1 x2 x x1 xx12 x2 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 Uniform Probability Distribution Example: Slater's Buffet Slater customers are charged for salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces. Uniform Probability Distribution f(x) = 1/10 for 5 < x < 15 =0 elsewhere where: x = salad plate filling weight Uniform Probability Distribution 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 Uniform Probability Distribution f(x) 1/10 0 5 10 15 Salad Weight (oz.) x Uniform Probability Distribution 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) = 0.3 1/10 0 5 10 12 Salad Weight (oz.) x 15 Area as a Measure of Probability The area under the graph of f(x) and probability are identical. This is valid for all continuous random variables. The probability that x takes on a value between some lower value x1 and some higher value x2 can be found by computing the area under the graph of f(x) over the interval from x1 to x2. Chapter 06 Continuous Probability Distributions 6.2 Normal Probability Distribution 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. It has been used in a wide variety of applications including: • Heights of people • Rainfall amounts • Test scores • Scientific measurements Abraham de Moivre, a French mathematician, published The Doctrine of Chances in 1733. He derived the normal distribution. Normal Probability Distribution Normal Probability Density Function 𝑓 𝑥 = 1 𝜎 2𝜋 2 /2𝜎2 −(𝑥−𝜇) 𝑒 where:  = mean  = standard deviation  = 3.14159 e = 2.71828 Normal Probability Distribution Characteristics The distribution is symmetric; its skewness measure is zero. x Normal Probability Distribution The entire family of normal probability distributions is defined by its mean  and its standard deviation  . Standard Deviation  Mean  x Normal Probability Distribution The highest point on the normal curve is at the mean, which is also the median and mode. x Normal Probability Distribution The mean can be any numerical value: negative, zero, or positive. x -10 0 25 Normal Probability Distribution The standard deviation determines the width of the curve: larger values result in wider, flatter curves.  = 15  = 25 x Normal Probability Distribution 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 (basis for the empirical rule) 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 (basis for the empirical rule) 99.72% 95.44% 68.26%  – 3  – 2  – 1   + 1  + 3  + 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. = z 0 Standard Normal Probability Distribution Converting to the Standard Normal Distribution 𝑥−𝜇 z= 𝜎 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 NORMSDIST is used to compute the cumulative 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 Excel 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) =NORM.S.DIST(1) =NORM.S.DIST(1)-NORM.S.DIST(0) =NORM.S.DIST(1.25)-NORM.S.DIST(0) =NORM.S.DIST(1)-NORM.S.DIST(-1) =1-NORM.S.DIST(1.58) =NORM.S.DIST(-0.5) Using Excel to Compute Standard Normal Probabilities Excel 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 Excel 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 =NORM.S.INV(0.9) =NORM.S.INV(0.975) =NORM.S.INV(0.025) Using Excel to Compute Standard Normal Probabilities Excel 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 Standard Normal Probability Distribution Example: Pep Zone Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed. The store manager is concerned that sales are being lost due to stockouts while waiting for a replenishment order. Standard Normal Probability Distribution It has been determined that demand during replenishment lead-time 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 during replenishment lead-time. In other words, what is the probability that demand during lead-time will exceed 20 gallons? P(x > 20) = ? Standard Normal Probability Distribution Solving for the Stockout Probability 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 to the left of z = .83. see next slide Standard Normal Probability Distribution Cumulative Probability Table for the Standard Normal Distribution z . .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 . . . . . . . . . . .5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190 .7224 .6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517 .7549 .7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823 .7852 .8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106 .8133 .9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365 .8389 . . . . . . . . . . . P(z < .83) Standard Normal Probability Distribution Solving for the Stockout Probability Step 3: Compute the area under the standard normal curve to the right of z = .83. P(z > .83) = 1 – P(z < .83) = 1- .7967 = .2033 Probability of a stockout P(x > 20) Standard Normal Probability Distribution Solving for the Stockout Probability Area = 1 - .7967 Area = .7967 = .2033 0 .83 z Standard Normal Probability Distribution Standard Normal Probability Distribution If the manager of Pep Zone wants the probability of a stockout during replenishment lead-time to be no more than .05, what should the reorder point be? --------------------------------------------------------------(Hint: Given a probability, we can use the standard normal table in an inverse fashion to find the corresponding z value.) Standard Normal Probability Distribution Solving for the Reorder Point Area = .9500 Area = .0500 0 z.05 z Standard Normal Probability Distribution 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 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429 .9441 1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535 .9545 1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625 .9633 1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 We .9693 look.9699 up .9706 complement 1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 the .9756 .9761 .9767 . . . . . . . . of the tail area . (1 -..05 = .95) . Standard Normal Probability Distribution Solving for the Reorder Point 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. Normal Probability Distribution Solving for the Reorder Point Probability of no Probability of a stockout during replenishment lead-time = .95 stockout during replenishment lead-time = .05 15 24.87 x Standard Normal Probability Distribution 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: NORM.DIST is used to compute the cumulative probability given an x value. NORM.INV is used to compute the x value given a cumulative probability. Using Excel to Compute Normal Probabilities Excel Formula Worksheet 1 2 3 4 5 6 7 8 A B Probabilities: Normal Distribution P (x > 20) =1-NORM.DIST(20,15,6,TRUE) Finding x Values, Given Probabilities x value with .05 in upper tail =NORM.INV(0.95,15,6) Using Excel to Compute Normal Probabilities Excel Formula Worksheet 1 2 3 4 5 6 7 8 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.