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Statistics

Statistics. Continuous Probability Distributions. Continuous Probability Distributions. Uniform Probability Distribution Normal Probability Distribution Normal Approximation of Binomial Probabilities Exponential Probability Distribution. Exponential. f ( x ). Uniform. f ( x ).

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Statistics

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  1. Statistics Continuous Probability Distributions

  2. Continuous Probability Distributions • Uniform Probability Distribution • Normal Probability Distribution • Normal Approximation of Binomial Probabilities • Exponential Probability Distribution

  3. Exponential f (x) Uniform f (x) Normal f (x) x x x Continuous Probability Distributions

  4. STATISTICSin PRACTICE • Procter & Gamble (P&G) produces and markets such products as detergents, disposable diapers, bar soaps, and paper towels. • The Industrial Chemicals Division of P&G is a supplier of fatty alcohols derived from natural substances such as coconut oil and from petroleum- based derivatives.

  5. STATISTICSin PRACTICE • The division wanted to know the economic risks and opportunities of expanding its fatty-alcohol production facilities, so it called in P&G’s experts in probabilistic decision and risk analysis to help.

  6. 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.

  7. 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 functionf(x) betweenx1and x2. • f(x) ≧0

  8. Exponential f (x) Uniform f (x) Normal f (x) x x1 x2 x1 x2 x x1 x2 x x1 x2 Continuous Probability Distributions

  9. Continuous Probability Distributions • The expected value of a continuous random variable x is: • The variance of a continuous random variable x is:

  10. Continuous Probability Distributions • If the random variable x has the density function f(x), the probability of the random variable assuming a value within some given interval from x1 to x2 is defined to be

  11. Continuous Probability Distributions • Cumulative probability function: • If the random variable x has the density function f(x), the cumulative distribution function for x ≦x2is:

  12. 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

  13. Uniform Probability Distribution • Expected Value of x E(x) = (a + b)/2 • Variance of x Var(x) = (b - a)2/12

  14. Uniform Probability Distribution • Example: Random variable x = the flight time of an airplane traveling from Chicago to New York. Suppose the flight time can be any value in the interval from 120 minutes to 140 minutes.

  15. Uniform Probability Distribution Assumeevery 1-minute interval being equally likely, x is said to have a uniform probability distributionand the probability density function is

  16. Uniform Probability Distribution • Example: Uniform Probability Density Function for Flight Time

  17. Uniform Probability Distribution • What is the probability that the flight time is between 120 and 130 minutes? That is, what is ? • Area provides Probability of Flight Time Between 120 and 130 Minutes

  18. Uniform Probability Distribution • Applying these formulas to the uniform distribution for flight times from Chicago to New York, we obtain and σ=5.77 minutes.

  19. 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 uniformly distributed between 5 ounces and 15 ounces.

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

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

  22. 5 10 15 Uniform Probability Distribution • Uniform Probability Distribution for Salad Plate Filling Weight f(x) 1/10 x Salad Weight (oz.)

  23. 5 10 12 15 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) = .3 1/10 x Salad Weight (oz.)

  24. 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.

  25. Normal Probability Distribution • It has been used in a wide variety of applications: Heights of people Scientific measurements

  26. Normal Probability Distribution • It has been used in a wide variety of applications: Test scores • Amounts • of rainfall

  27.  μ= mean, σ = standard deviation, π = 3.14159 e = 2.71828 Normal Probability Distribution • Normal Probability Density Function where:

  28. Normal Probability Distribution • Characteristics The distribution is symmetric; its skewness measure is zero. x

  29. Normal Probability Distribution • Characteristics The entire family of normal probability distributions is defined by itsmeanμand its standard deviationσ. Standard Deviation s x Mean m

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

  31. Normal Probability Distribution • Characteristics The mean can be any numerical value: negative, zero, or positive. x -10 0 20

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

  33. 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

  34. of values of a normal random variable • are within of its mean. 68.26% +/- 1 standard deviation • of values of a normal random variable • are within of its mean. 95.44% +/- 2 standard deviations • of values of a normal random variable • are within of its mean. 99.72% +/- 3 standard deviations Normal Probability Distribution • Characteristics

  35. 99.72% 95.44% 68.26% Normal Probability Distribution • Characteristics x m m + 3s m – 3s m – 1s m + 1s m – 2s m + 2s

  36. 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.

  37. Standard Normal Probability Distribution • The letter z is used to designate the standard • normal random variable. s = 1 z 0

  38. Standard Normal Probability Distribution • Areas, or probabilities, for The Standard Normal Distribution

  39. Standard Normal Probability Distribution

  40. Standard Normal Probability Distribution • Example: What is the probability that the z value for the standard normal random variable will be between .00 and 1.00?

  41. Standard Normal Probability Distribution

  42. Standard Normal Probability Distribution • Example: = ?

  43. Standard Normal Probability Distribution • Example: = ? • Table 6.1 to show that the probability of a z value between z = .00 and z = 1.00 is .3413 • the normal distribution is symmetric, therefore, = = .3413 + .3413 = .6826

  44. Standard Normal Probability Distribution • Example: = ? • = the normal distribution is symmetric and = = .1915 + .5000 = .6915.

  45. Standard Normal Probability Distribution

  46. Standard Normal Probability Distribution • Example: = ? • Probability of a z value between z = 0.00 and z = 1.00 is .3413, and Probability of a z value between z = 0.00 and z =1.58 is .4429 . Hence, Probability of a z value between z = 1.00 and z = 1.58 is .4429 — .3413 = .1016.

  47. Standard Normal Probability Distribution

  48. Standard Normal Probability Distribution • Example: find a z value such that the probability of obtaining a larger z value is .10.

  49. Standard Normal Probability Distribution • An area of approximately .4000 (actually .3997) will be between the mean and z 1.28.* In terms of the question originally asked, the probability is approximately .10 that the z value will be larger than 1.28.

  50. Standard Normal Probability Distribution • Converting to the Standard Normal Distribution We can think ofzas a measure of the number of standard deviationsxis from.

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