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1 1 Slide Continuous Probability Distributions Chapter 6 BA 201

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2 2 Slide Continuous Probability Distributions n n A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. n n It is not possible to talk about the probability of the random variable assuming a particular value. n n Instead, we talk about the probability of the random variable assuming a value within a given interval.

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3 3 Slide Continuous Probability Distributions n n The probability of the random variable assuming a value within some given interval from x 1 to x 2 is defined to be the area under the graph of the probability density function between x 1 and x 2. f ( x ) x x Uniform x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 x Normal x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 Exponential x x x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2

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4 4 Slide Area as a Measure of Probability n n The area under the graph of f ( x ) and probability are identical. n n This is valid for all continuous random variables. n n The probability that x takes on a value between some lower value x 1 and some higher value x 2 can be found by computing the area under the graph of f ( x ) over the interval from x 1 to x 2.

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5 5 Slide UNIFORM PROBABILITY DISTRIBUTION

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6 6 Slide Uniform Probability Distribution where: a = smallest value the variable can assume b = largest value the variable can assume f ( x ) = 1/( b – a ) for a < x < b = 0 elsewhere n n A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. n n The uniform probability density function is: f ( x ) x x Uniform x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2

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7 7 Slide Var( x ) = ( b - a ) 2 /12 E( x ) = ( a + b )/2 Uniform Probability Distribution n Expected Value of x n Variance of x

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8 8 Slide Uniform Probability Distribution Slater's Buffet 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.

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9 9 Slide n Uniform Probability Density Function f ( x ) = 1/10 for 5 < x < 15 = 0 elsewhere where: x = salad plate filling weight Uniform Probability Distribution

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10 Slide n Expected Value of x E( x ) = ( a + b )/2 = (5 + 15)/2 = 10 Var( x ) = ( b - a ) 2 /12 = (15 – 5) 2 /12 = 8.33 Uniform Probability Distribution n Variance of x

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11 Slide n Uniform Probability Distribution for Salad Plate Filling Weight f(x)f(x) f(x)f(x) x x 1/10 Salad Weight (oz.) Uniform Probability Distribution

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12 Slide f(x)f(x) f(x)f(x) x x 1/10 Salad Weight (oz.) P(12 < x < 15) = 1/10(3) =.3 What is the probability that a customer will take between 12 and 15 ounces of salad? Uniform Probability Distribution 12

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13 Slide UNIFORM PROBABILITY DISTRIBUTION PRACTICE

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14 Slide Scenario Smile time in eight week old babies ranges from zero to 23 seconds What is f(x)? Compute E(x) Compute Var(x) What is the probability a baby smiles a. a. Less than or equal to five seconds? b. b. 10 to 18 seconds inclusive?

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15 Slide NORMAL PROBABILITY DISTRIBUTION

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16 Slide Normal Probability Distribution n n The normal probability distribution is the most important distribution for describing a continuous random variable. n n It is widely used in statistical inference. n n It has been used in a wide variety of applications including: Heights of people Rainfall amounts Test scores Scientific measurements

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17 Slide The distribution is symmetric; its skewness measure is zero. Normal Probability Distribution n Characteristics x

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18 Slide The entire family of normal probability distributions is defined by its mean and its standard deviation . Normal Probability Distribution n Characteristics Standard Deviation Mean x

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19 Slide The highest point on the normal curve is at the mean, which is also the median and mode. Normal Probability Distribution n Characteristics x

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20 Slide Normal Probability Distribution n Characteristics The mean can be any numerical value: negative, zero, or positive. x

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21 Slide Normal Probability Distribution n Characteristics = 15 The standard deviation determines the width of the curve: larger values result in wider, flatter curves. = 25 x

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22 Slide Probabilities for the normal random variable are given by areas under the curve. The total area under the curve is 1 (0.5 to the left of the mean and 0.5 to the right). Normal Probability Distribution n Characteristics.5.5 x

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23 Slide Normal Probability Distribution n Characteristics (basis for the empirical rule) x – 1 + 1 68.26% – 2 + 2 95.44% – 3 + 3 99.72%

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24 Slide Normal Probability Distribution n Normal Probability Density Function = mean = standard deviation = e = where:

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25 Slide NORMAL PROBABILITY DISTRIBUTION PRACTICE

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26 Slide Scenario Mean75pi Std Dev8.95e Compute f(75), f(66), and f(84)

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27 Slide STANDARD NORMAL PROBABILITY DISTRIBUTION

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28 Slide 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. n Characteristics

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29 Slide 0 z The letter z is used to designate the standard normal random variable. Standard Normal Probability Distribution n Characteristics

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30 Slide n Converting to the Standard Normal Distribution Standard Normal Probability Distribution We can think of z as a measure of the number of standard deviations x is from .

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31 Slide Standard Normal Probability Distribution 0 z P(z ≤ 0)=0.500 z=0 P(z ≤ -1)= z=-1 P(z ≤ -2)=0.023 z=-2

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32 Slide Standard Normal Probability Distribution 0 z P(z ≤ 1)= z=1

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33 Slide Standard Normal Probability Distribution 0 z P(z ≤ 1)= z=1 P(z ≤ -1)= z=-1 P(-1 ≤ z ≤ +1) P(z ≤ 1) P(z ≤ -1) == 0.682=

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34 Slide Standard Normal Probability Distribution 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.

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35 Slide 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. Standard Normal Probability Distribution Pep Zone 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) = ?

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36 Slide z = ( x - )/ = ( )/6 = 0.83 n Solving for the Stockout Probability Step 1: Convert x to the standard normal distribution. Step 2: Find the area under the standard normal curve to the left of z = Standard Normal Probability Distribution

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37 Slide n Cumulative Probability Table for the Standard Normal Distribution Standard Normal Probability Distribution z P ( z < 0.83) Appendix B Table 1

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38 Slide P ( z > 0.83) = 1 – P ( z < 0.83) = = n Solving for the Stockout Probability Step 3: Compute the area under the standard normal curve to the right of z = Probability of a stockout P ( x > 20) Standard Normal Probability Distribution

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39 Slide n Solving for the Stockout Probability Area = Area = = z Standard Normal Probability Distribution

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40 Slide n 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 0.05, 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.)

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41 Slide n Solving for the Reorder Point 0 Area = Area = z z 0.05 Standard Normal Probability Distribution

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42 Slide n Solving for the Reorder Point Step 1: Find the z -value that cuts off an area of 0.05 in the right tail of the standard normal distribution. We look up the complement of the tail area ( = 0.95) Standard Normal Probability Distribution

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43 Slide n Solving for the Reorder Point Step 2: Convert z 0.05 to the corresponding value of x. x = + z 0.05 = (6) = or 25 A reorder point of 25 gallons will place the probability of a stockout during leadtime at (slightly less than) Standard Normal Probability Distribution

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44 Slide Normal Probability Distribution n Solving for the Reorder Point 15 x Probability of a stockout during replenishment lead-time = 0.05 Probability of no stockout during replenishment lead-time = 0.95

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45 Slide n 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. Standard Normal Probability Distribution

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46 Slide STANDARD NORMAL PROBABILITY DISTRIBUTION PRACTICE

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47 Slide Scenario (6-20)

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48 Slide a) P(x<=50)

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49 Slide Scenario (6-22)

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50 Slide

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