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 4 Continuous Random Variables and Probability Distributions Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 4.1 Continuous Random Variables and Probability Distributions Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Continuous Random Variables A random variable X is continuous if its set of possible values is an entire interval of numbers (If A < B, then any number x between A and B is possible). Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Probability Distribution Let X be a continuous rv. Then a probability distribution or probability density function (pdf) of X is a function f (x) such that for any two numbers a and b, P a X b f ( x)dx b a The graph of f is the density curve. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Probability Density Function For f (x) to be a pdf 1. f (x) > 0 for all values of x. 2.The area of the region between the graph of f and the x – axis is equal to 1. y f ( x) Area = 1 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Probability Density Function P(a X b) is given by the area of the shaded region. y f ( x) a b Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Uniform Distribution A continuous rv X is said to have a uniform distribution on the interval [A, B] if the pdf of X is 1 A x B f x; A, B B A 0 otherwise Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Probability for a Continuous rv If X is a continuous rv, then for any number c, P(x = c) = 0. For any two numbers a and b with a < b, P ( a X b) P ( a X b) P ( a X b) P ( a X b) Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 4.2 Cumulative Distribution Functions and Expected Values Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Cumulative Distribution Function The cumulative distribution function, F(x) for a continuous rv X is defined for every number x by F ( x) P X x f ( y)dy x For each x, F(x) is the area under the density curve to the left of x. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Using F(x) to Compute Probabilities Let X be a continuous rv with pdf f(x) and cdf F(x). Then for any number a, P X a 1 F (a) and for any numbers a and b with a < b, P a X b F (b) F (a) Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Obtaining f(x) from F(x) If X is a continuous rv with pdf f(x) and cdf F(x), then at every number x for which the derivative F ( x) exists, F ( x) f ( x). Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Percentiles Let p be a number between 0 and 1. The (100p)th percentile of the distribution of a continuous rv X denoted by ( p ), is defined by p F ( p) ( p) f ( y)dy Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Median The median of a continuous distribution, denoted by , is the 50th percentile. So satisfies 0.5 F ( ). That is, half the area under the density curve is to the left of . Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Expected Value The expected or mean value of a continuous rv X with pdf f (x) is X E X x f ( x)dx Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Expected Value of h(X) If X is a continuous rv with pdf f(x) and h(x) is any function of X, then E h( x ) h ( X ) h( x) f ( x)dx Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Variance and Standard Deviation The variance of continuous rv X with pdf f(x) and mean is 2 X V ( x) (x ) 2 f ( x)dx E[ X ] 2 The standard deviation is X V ( x). Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Short-cut Formula for Variance E ( X ) V (X ) E X 2 2 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 4.3 The Normal Distribution Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Normal Distributions A continuous rv X is said to have a normal distribution with parameters and , where and 0 , if the pdf of X is 1 ( x )2 /(2 2 ) f ( x) e 2 x Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Standard Normal Distributions The normal distribution with parameter values 0 and 1 is called a standard normal distribution. The random variable is denoted by Z. The pdf is 1 z2 / 2 f ( z;0,1) e z 2 The cdf is z ( z ) P( Z z ) f ( y;0,1)dy Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Standard Normal Cumulative Areas Shaded area = (z ) Standard normal curve 0 z Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Standard Normal Distribution Let Z be the standard normal variable. Find (from table) a. P( Z 0.85) Area to the left of 0.85 = 0.8023 b. P(Z > 1.32) 1 P( Z 1.32) 0.0934 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. c. P(2.1 Z 1.78) Find the area to the left of 1.78 then subtract the area to the left of –2.1. = P( Z 1.78) P( Z 2.1) = 0.9625 – 0.0179 = 0.9446 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. z Notation z will denote the value on the measurement axis for which the area under the z curve lies to the right of z . Shaded area P(Z z ) 0 z Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Let Z be the standard normal variable. Find z if a. P(Z < z) = 0.9278. Look at the table and find an entry = 0.9278 then read back to find z = 1.46. b. P(–z < Z < z) = 0.8132 P(z < Z < –z ) = 2P(0 < Z < z) = 2[P(z < Z ) – ½] = 2P(z < Z ) – 1 = 0.8132 P(z < Z ) = 0.9066 z = 1.32 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Nonstandard Normal Distributions If X has a normal distribution with mean and standard deviation , then Z X has a standard normal distribution. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Normal Curve Approximate percentage of area within given standard deviations (empirical rule). 99.7% 95% 68% Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Let X be a normal random variable with 80 and 20. Find P( X 65). 65 80 P X 65 P Z 20 P Z .75 = 0.2266 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Ex. A particular rash shown up at an elementary school. It has been determined that the length of time that the rash will last is normally distributed with 6 days and 1.5 days. Find the probability that for a student selected at random, the rash will last for between 3.75 and 9 days. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 96 3.75 6 P 3.75 X 9 P Z 1.5 1.5 P 1.5 Z 2 = 0.9772 – 0.0668 = 0.9104 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Percentiles of an Arbitrary Normal Distribution (100p)th percentile (100 p)th for , for normal standard normal Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Normal Approximation to the Binomial Distribution Let X be a binomial rv based on n trials, each with probability of success p. If the binomial probability histogram is not too skewed, X may be approximated by a normal distribution with np and npq . x 0.5 np P( X x) npq Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Ex. At a particular small college the pass rate of Intermediate Algebra is 72%. If 500 students enroll in a semester determine the probability that at least 375 students pass. np 500(.72) 360 npq 500(.72)(.28) 10 375.5 360 P( X 375) (1.55) 10 = 0.9394 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 4.4 The Gamma Distribution and Its Relatives Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Gamma Function For 0, the gamma function ( ) is defined by 1 x ( ) x e dx 0 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Gamma Distribution A continuous rv X has a gamma distribution if the pdf is 1 1 x / x e x0 f ( x; , ) ( ) 0 otherwise where the parameters satisfy 0, 0. The standard gamma distribution has 1. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Mean and Variance The mean and variance of a random variable X having the gamma distribution f ( x; , ) are E( X ) V ( X ) 2 2 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Probabilities from the Gamma Distribution Let X have a gamma distribution with parameters and . Then for any x > 0, the cdf of X is given by x P( X x) F ( x; , ) F ; where x F ( x; ) 0 1 y y e ( ) dy Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Exponential Distribution A continuous rv X has an exponential distribution with parameter if the pdf is e x x 0 f ( x; ) 0 otherwise Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Mean and Variance The mean and variance of a random variable X having the exponential distribution 1 2 2 1 2 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Probabilities from the Gamma Distribution Let X have a exponential distribution Then the cdf of X is given by x0 0 F ( x; ) x x0 1 e Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Applications of the Exponential Distribution Suppose that the number of events occurring in any time interval of length t has a Poisson distribution with parameter t and that the numbers of occurrences in nonoverlapping intervals are independent of one another. Then the distribution of elapsed time between the occurrences of two successive events is exponential with parameter . Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Chi-Squared Distribution Let v be a positive integer. Then a random variable X is said to have a chisquared distribution with parameter v if the pdf of X is the gamma density with v / 2 and 2. The pdf is 1 (v / 2)1 x / 2 x e v/2 f ( x; v) 2 (v / 2) 0 x0 x0 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Chi-Squared Distribution The parameter v is called the number of degrees of freedom (df) of X. The 2 symbol is often used in place of “chisquared.” Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 4.5 Other Continuous Distributions Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Weibull Distribution A continuous rv X has a Weibull distribution if the pdf is 1 ( x / ) x e f ( x; , ) 0 x0 x0 where the parameters satisfy 0, 0. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Mean and Variance The mean and variance of a random variable X having the Weibull distribution are 2 1 2 1 2 2 1 1 1 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Weibull Distribution The cdf of a Weibull rv having parameters and is 1 e F ( x; , ) ( x / ) 0 x0 x<0 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Lognormal Distribution A nonnegative rv X has a lognormal distribution if the rv Y = ln(X) has a normal distribution the resulting pdf has parameters and and is 1 [ln( x ) ]2 /(2 2 ) e f ( x; , ) 2 x 0 x0 x0 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Mean and Variance The mean and variance of a variable X having the lognormal distribution are E( X ) e 2 / 2 V (X ) e 2 2 e 2 1 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Lognormal Distribution The cdf of the lognormal distribution is given by F ( x; , ) P( X x) P[ln( X ) ln( x)] ln( x) ln( x) PZ Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Beta Distribution A rv X is said to have a beta distribution with parameters A, B, 0, and 0 if the pdf of X is f ( x; , , A, B) 1 1 1 ( ) x A B x B A ( ) ( ) B A B A 0 otherwise x0 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Mean and Variance The mean and variance of a variable X having the beta distribution are A ( B A) 2 ( B A) 2 ( ) ( 1) 2 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 4.6 Probability Plots Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Sample Percentile Order the n-sample observations from smallest to largest. The ith smallest observation in the list is taken to be the [100(i – 0.5)/n]th sample percentile. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Probability Plot [100(i .5) / n]th percentile ith smallest sample observation of the distribution , If the sample percentiles are close to the corresponding population distribution percentiles, the first number will roughly equal the second. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Normal Probability Plot A plot of the pairs [100(i .5) / n]th z percentile, ith smallest observation On a two-dimensional coordinate system is called a normal probability plot. If the drawn from a normal distribution the points should fall close to a line with slope and intercept . Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Beyond Normality Consider a family of probability distributions involving two parameters 1 and 2 . Let F ( x;1,2 ) denote the corresponding cdf’s. The parameters 1 and 2 are said to location and scale parameters if x 1 F ( x;1, 2 ) is a function of . 2 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.