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1 1 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Chapter-6 Continuous Probability Distributions

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2 2 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Learning Objectives n Identify Types of Continuous Probability Distribution and Learn their respective Functional Representation (Formulas) n Learn how to calculate the Mean and Variance of a random Variable that assumes each o type of Continuous Probability Distribution Identified.

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3 3 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Continuous Probability Distributions n Uniform Probability Distribution n Normal Probability Distribution n Exponential Probability Distribution f ( x ) x x Uniform x Normal x x Exponential

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4 4 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Continuous Probability Distributions n A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. n It is not possible to talk about the probability of the random variable assuming a particular value. n Instead, we talk about the probability of the random variable assuming a value within a given interval.

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5 5 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Continuous Probability Distributions 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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6 6 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 6.1) Uniform Continuous Probability Distribution where: a = smallest value the variable can assume b = largest value the variable can assume b = largest value the variable can assume f ( x ) = 1/( b – a ) for a < x < b f ( x ) = 1/( b – a ) for a < x < b = 0 elsewhere = 0 elsewhere f ( x ) = 1/( b – a ) for a < x < b f ( x ) = 1/( b – a ) for a < x < b = 0 elsewhere = 0 elsewhere n A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. n The uniform probability density function is:

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7 7 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Var ( x ) = ( b - a ) 2 /12 E( x ) = ( a + b )/2 6.1) Uniform Probability Distribution n Expected Value of x n Variance of x

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8 8 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 6.1) Uniform Probability Distribution n Example: 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 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Uniform Probability Density Function f ( x ) = 1/10 for 5 < x < 15 f ( x ) = 1/10 for 5 < x < 15 = 0 elsewhere = 0 elsewhere f ( x ) = 1/10 for 5 < x < 15 f ( x ) = 1/10 for 5 < x < 15 = 0 elsewhere = 0 elsewhere where: x = salad plate filling weight x = salad plate filling weight 6.1) Uniform Probability Distribution

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10 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Expected Value of x n Variance of x E( x ) = ( a + b )/2 E( x ) = ( a + b )/2 = (5 + 15)/2 = (5 + 15)/2 = 10 = 10 E( x ) = ( a + b )/2 E( x ) = ( a + b )/2 = (5 + 15)/2 = (5 + 15)/2 = 10 = 10 Var( x ) = ( b - a ) 2 /12 Var( x ) = ( b - a ) 2 /12 = (15 – 5) 2 /12 = (15 – 5) 2 /12 = 8.33 = 8.33 Var( x ) = ( b - a ) 2 /12 Var( x ) = ( b - a ) 2 /12 = (15 – 5) 2 /12 = (15 – 5) 2 /12 = 8.33 = 8.33 Uniform Probability Distribution

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11 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Uniform Probability Distribution for Salad Plate Filling Weight f(x)f(x) f(x)f(x) x x 5 5 10 15 1/10 Salad Weight (oz.) 6.1) Uniform Probability Distribution

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12 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) f(x)f(x) f(x)f(x) x x 5 5 10 15 1/10 Salad Weight (oz.) P(12 < x < 15) = 1/10(3) =.3 What is the probability that a customer What is the probability that a customer will take between 12 and 15 ounces of salad? will take between 12 and 15 ounces of salad? 12 6.1) Uniform Probability Distribution

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13 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Normal Probability Distribution

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14 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 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

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15 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Heights of people Heights Normal Probability Distribution n It has been used in a wide variety of applications: Scientific measurements measurementsScientific

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16 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Amounts of rainfall Amounts Normal Probability Distribution n It has been used in a wide variety of applications: Test scores scoresTest

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17 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Normal Probability Distribution n Normal Probability Density Function = mean = standard deviation = 3.14159 e = 2.71828 where:

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18 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The distribution is symmetric; its skewness The distribution is symmetric; its skewness measure is zero. measure is zero. The distribution is symmetric; its skewness The distribution is symmetric; its skewness measure is zero. measure is zero. Normal Probability Distribution n Characteristics x

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19 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The entire family of normal probability The entire family of normal probability distributions is defined by its mean and its distributions is defined by its mean and its standard deviation . standard deviation . The entire family of normal probability The entire family of normal probability distributions is defined by its mean and its distributions is defined by its mean and its standard deviation . standard deviation . Normal Probability Distribution n Characteristics Standard Deviation Mean x

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20 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. Normal Probability Distribution n Characteristics x

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21 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Normal Probability Distribution n Characteristics -10020 The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. zero, or positive. The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. zero, or positive. x

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22 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Normal Probability Distribution n Characteristics = 15 = 25 The standard deviation determines the width of the curve: larger values result in wider, flatter curves. The standard deviation determines the width of the curve: larger values result in wider, flatter curves. x

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23 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Normal Probability Distribution n Characteristics.5.5 x

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24 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Normal Probability Distribution n Characteristics of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean.68.26%68.26% +/- 1 standard deviation of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. 95.44%95.44% +/- 2 standard deviations of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean. of values of a normal random variable of values of a normal random variable are within of its mean. are within of its mean.99.72%99.72% +/- 3 standard deviations

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25 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Normal Probability Distribution n Characteristics x – 3 – 1 – 2 + 1 + 2 + 3 68.26% 95.44% 99.72%

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26 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) STANDARD NORMAL PROBABILITY DISTRIBUTION STANDARD NORMAL PROBABILITY DISTRIBUTION A random variable that has a normal distribution A random variable that has a normal distribution and a mean of 0 and a standard deviation of 1 is and a mean of 0 and a standard deviation of 1 is called a standard normal probability distribution. called a standard normal probability distribution. A random variable that has a normal distribution A random variable that has a normal distribution and a mean of 0 and a standard deviation of 1 is and a mean of 0 and a standard deviation of 1 is called a standard normal probability distribution. called a standard normal probability distribution. A Special Type of Normal Distribution

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27 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 0 z The letter z is used to designate the standard The letter z is used to designate the standard normal random variable. normal random variable. The letter z is used to designate the standard The letter z is used to designate the standard normal random variable. normal random variable. Standard Normal Probability Distribution

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28 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) 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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29 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Standard Normal Probability Distribution n Standard Normal Density Function z = ( x – )/ = 3.14159 e = 2.71828 where:

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30 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Examples: Hands-on-Practice Problems

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31 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Standard Normal Probability Distribution n 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. Pep Zone 5w-20 Motor Oil

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32 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) The store manager is concerned that sales are being 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 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, P ( x > 20). Standard Normal Probability Distribution Pep Zone 5w-20 Motor Oil n Example: Pep Zone

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33 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) z = ( x - )/ z = ( x - )/ = (20 - 15)/6 = (20 - 15)/6 =.83 =.83 z = ( x - )/ z = ( x - )/ = (20 - 15)/6 = (20 - 15)/6 =.83 =.83 n Solving for the Stockout Probability Step 1: Convert x to the standard normal distribution. Pep Zone 5w-20 Motor Oil Step 2: Find the area under the standard normal curve to the left of z =.83. curve to the left of z =.83. Step 2: Find the area under the standard normal curve to the left of z =.83. curve to the left of z =.83. see next slide see next slide Standard Normal Probability Distribution

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34 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Cumulative Probability Table for the Standard Normal Distribution Pep Zone 5w-20 Motor Oil P ( z <.83) Standard Normal Probability Distribution

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35 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) P ( z >.83) = 1 – P ( z.83) = 1 – P ( z <.83) = 1-.7967 = 1-.7967 =.2033 =.2033 P ( z >.83) = 1 – P ( z.83) = 1 – P ( z <.83) = 1-.7967 = 1-.7967 =.2033 =.2033 n Solving for the Stockout Probability Step 3: Compute the area under the standard normal curve to the right of z =.83. curve to the right of z =.83. Step 3: Compute the area under the standard normal curve to the right of z =.83. curve to the right of z =.83. Pep Zone 5w-20 Motor Oil Probability of a stockout of a stockout P ( x > 20) Standard Normal Probability Distribution

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36 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Solving for the Stockout Probability 0.83 Area =.7967 Area = 1 -.7967 =.2033 =.2033 z Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution

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37 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n 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? Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution

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38 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil 0 Area =.9500 Area =.0500 z z.05 Standard Normal Probability Distribution

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39 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil Step 1: Find the z -value that cuts off an area of.05 in the right tail of the standard normal in the right tail of the standard normal distribution. distribution. Step 1: Find the z -value that cuts off an area of.05 in the right tail of the standard normal in the right tail of the standard normal distribution. distribution. We look up the complement of the tail area (1 -.05 =.95) Standard Normal Probability Distribution

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40 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil Step 2: Convert z.05 to the corresponding value of x. x = + z.05 x = + z.05 = 15 + 1.645(6) = 24.87 or 25 = 24.87 or 25 x = + z.05 x = + z.05 = 15 + 1.645(6) = 24.87 or 25 = 24.87 or 25 A reorder point of 25 gallons will place the probability A reorder point of 25 gallons will place the probability of a stockout during leadtime at (slightly less than).05. of a stockout during leadtime at (slightly less than).05. Standard Normal Probability Distribution

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41 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Solving for the Reorder Point Pep Zone 5w-20 Motor Oil By raising the reorder point from 20 gallons to 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 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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42 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Normal Approximation of Binomial Probabilities

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43 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Normal Approximation of Binomial Probabilities When the number of trials, n, becomes large, When the number of trials, n, becomes large, we can use the normal distribution to approximate the binomial probability distribution That is, the normal probability distribution also serves as a means to approximation of binomial probabilities: That is, the normal probability distribution also serves as a means to approximation of binomial probabilities: Pre-conditions : Pre-conditions : n > 20, np > 5, and n (1 - p ) > 5. n > 20, np > 5, and n (1 - p ) > 5.

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44 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Normal Approximation of Binomial Probabilities Set = np Set = np n When using the normal distribution to approximate binomial probabilities, however, we add and a subtract 0.5 as a correction factor. n That is, 0.5 is used as a continuity correction factor because a continuous distribution is being used to approximate a discrete distribution. n For example, P ( x = 10) is approximated by P (9.5 < x < 10.5).

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45 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Exponential Probability Distribution

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46 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Exponential Probability Distribution n Often times, the exponential probability distribution describes the time it takes to complete a task. n The exponential random variables can be used to describe: Time between vehicle arrivals at a toll booth Time between vehicle arrivals at a toll booth Time required to complete a questionnaire Time required to complete a questionnaire Distance between major defects in a highway Distance between major defects in a highway SLOW

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47 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Density Function Exponential Probability Distribution where: = mean e = 2.71828 e = 2.71828 for x > 0, > 0

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48 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Exponential Probability Distribution where: x 0 = some specific value of x x 0 = some specific value of x Cumulative Probabilities Cumulative Probabilities

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49 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Exponential Probability Distribution The exponential distribution is skewed to the right. The exponential distribution is skewed to the right. The skewness measure for the exponential distribution The skewness measure for the exponential distribution is 2. is 2. The skewness measure for the exponential distribution The skewness measure for the exponential distribution is 2. is 2. x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 Exponential x x f ( x ) x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2

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50 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Relationship between the Poisson and Exponential Distributions The Poisson distribution provides an appropriate description of the number of occurrences per interval 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 The exponential distribution provides an appropriate description of the length of the interval between occurrences x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2 Exponential x x f ( x ) x1 x1x1 x1 x1 x1x1 x1 x2 x2x2 x2 x2 x2x2 x2

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51 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) n Hands-on-Practice Problem

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52 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Exponential Probability Distribution n Example: Al’s Full-Service Pump The time between arrivals of cars 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.

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53 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) x x f(x)f(x) f(x)f(x).1.3.4.2 1 2 3 4 5 6 7 8 9 10 Time Between Successive Arrivals (mins.) Exponential Probability Distribution P ( x < 2) = 1 - 2.71828 -2/3 = 1 -.5134 =.4866 P ( x < 2) = 1 - 2.71828 -2/3 = 1 -.5134 =.4866

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54 Slide 2009 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Exponential Probability Distribution A property of the exponential distribution is that A property of the exponential distribution is that the mean, , and standard deviation, , are equal. the mean, , and standard deviation, , are equal. A property of the exponential distribution is that A property of the exponential distribution is that the mean, , and standard deviation, , are equal. the mean, , and standard deviation, , are equal. Thus, the standard deviation, , and variance, 2, for the time between arrivals at Al’s full-service pump are: = = 3 minutes 2 = (3) 2 = 9

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