Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter-6 Continuous Probability Distributions University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 1 Learning Objectives Identify the different types of Continuous Probability Distribution and Learn their respective Functional Representation (Formulas) Learn how to calculate the Mean and Variance of a random Variable that assumes each o type of Continuous Probability Distribution Identified. University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 2 Continuous Probability Distributions Uniform Probability Distribution Normal Probability Distribution Exponential Probability Distribution f (x) f (x) Exponential x Uniform f (x) Normal x x University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 3 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. University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 4 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 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 5 6.1) Uniform Continuous 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 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 6 6.1) Uniform Probability Distribution Expected Value of x E(x) = (a + b)/2 Variance of x Var (x) = (b - a)2/12 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 7 6.1) Uniform Probability Distribution Example: Slater's Buffet Slater customers are charged for the amount of salad they take. Sampling suggests that the amount of salad taken is by customers is uniformly distributed between 5 ounces and 15 ounces. University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 8 6.1) Uniform Probability Distribution Uniform Probability Density Function f(x) = 1/10 for 5 < x < 15 =0 elsewhere where: x = salad plate filling weight University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 9 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 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 10 Normal Probability Distribution University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 13 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. University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 14 Normal Probability Distribution It has been used in a wide variety of applications: Heights of people Scientific measurements University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 15 Normal Probability Distribution It has been used in a wide variety of applications: Test scores Amounts of rainfall University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 16 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 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 17 Normal Probability Distribution Characteristics The distribution is symmetric; its skewness measure is zero. x University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 18 Normal Probability Distribution Characteristics The entire family of normal probability distributions is defined by its mean and its standard deviation . Standard Deviation Mean x University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 19 Normal Probability Distribution Characteristics The highest point on the normal curve is at the mean, which is also the median and mode. x University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 20 Normal Probability Distribution Characteristics The mean can be any numerical value: negative, zero, or positive. x -10 0 20 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 21 Normal Probability Distribution Characteristics The standard deviation determines the width of the curve: larger values result in wider, flatter curves. = 15 = 25 x University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 22 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 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 23 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. University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 24 Normal Probability Distribution Characteristics 99.72% 95.44% 68.26% – 3 – 1 – 2 + 3 + 1 + 2 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) x Slide 25 A Special Type of Normal Distribution STANDARD NORMAL PROBABILITY DISTRIBUTION A random variable that has a normal distribution and a mean of 0 and a standard deviation of 1 is called a standard normal probability distribution. University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 26 Standard Normal Probability Distribution The letter z is used to designate the standard normal random variable. 1 z 0 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 27 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 . University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 28 Standard Normal Probability Distribution Standard Normal Density Function 1 z2 /2 f (x) e 2 where: z = (x – )/ = 3.14159 e = 2.71828 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 29 Examples: Hands-on-Practice Problems University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 30 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. University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Pep Zone 5w-20 Motor Oil Slide 31 Standard Normal Probability Distribution Example: Pep Zone 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. Pep Zone 5w-20 Motor Oil The manager would like to know the probability of a stockout, P(x > 20). University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 32 Standard Normal Probability Distribution Pep Zone 5w-20 Motor Oil Solving for the Stockout Probability Step 1: Formulate the question in a mathematical format P (x > 20)? University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 33 Standard Normal Probability Distribution Pep Zone 5w-20 Motor Oil Solving for the Stockout Probability Step 2: Convert x to the standard normal distribution ( that is, standardize the random variable). z = (x - )/ = (20 - 15)/6 = .83 Step 3: Then find the area under the standard normal curve to the left of z = .83 (that is. P(x < 20) = P(Z<.83) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 34 Standard Normal Probability Distribution Pep Zone 5w-20 Motor Oil 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) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 35 Standard Normal Probability Distribution Pep Zone Solving for the Stockout Probability 5w-20 Motor Oil Step 4: 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) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 36 Standard Normal Probability Distribution Pep Zone 5w-20 Motor Oil Solving for the Stockout Probability Area = 1 - .7967 Area = .7967 = .2033 0 .83 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) z Slide 37 Normal Approximation of Binomial Probabilities University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 43 Normal Approximation of Binomial Probabilities The normal probability distribution also serves as a means for approximating binomial probabilities: Pre-conditions : n > 20, np > 5, and n(1 - p) > 5. That is, when the number of trials, n, becomes large, we can use the normal distribution to approximate the binomial probability distribution University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 44 Normal Approximation of Binomial Probabilities Set = np np(1 p) When using the normal distribution to approximate binomial probabilities, however, we add and a subtract 0.5 as a correction factor. That is, 0.5 is used as a continuity correction factor because a continuous distribution is being used to approximate a discrete distribution. For example, P(x = 10) is approximated by P(9.5 < x < 10.5). University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 45 Exponential Probability Distribution University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 46 Exponential Probability Distribution Often times, the exponential probability distribution describes 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 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 47 Exponential Probability Distribution Density Function f ( x) 1 e x / for x > 0, > 0 where: = mean e = 2.71828 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 48 Exponential Probability Distribution Cumulative Probabilities P ( x x0 ) 1 e xo / where: x0 = some specific value of x University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 49 Exponential Probability Distribution The exponential distribution is skewed to the right. f (x) Exponential x1 xx12 x2 x The skewness measure for the exponential distribution is 2. University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 50 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 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Exponential f (x) x xx12x2 1 Slide 51 x Hands-on-Practice Problem University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 52 Exponential Probability Distribution Example: Al’s Full-Service Pump 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. University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 53 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.) University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 54 Exponential Probability Distribution A property of the exponential distribution is that 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 University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 55 A marketing research done by a fast food restaurant indicates that coffee tastes best if its temperature is between 153 and 167 degrees. The restaurant samples the coffee it serves and observes 24 temperature readings over a day. The readings have a mean of 160.083 and standard deviation of 5.372 degrees. Assuming that the readings are normally distributed, what is the probability that a randomly selected cup of coffee is outside the requirement for best testing coffee? University of Minnesota-Duluth, Econ-2030 (Dr. Tadesse) Slide 56