Probability Densities Chapter 5 Probability Densities Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Continuous Random Variables Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Random Variable For a given sample space S of some experiment, a random variable is any rule that associates a number with each outcome in S . Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Types of Random Variables A discrete random variable is an rv whose possible values either constitute a finite set or else can listed in an infinite sequence. A random variable is continuous if its set of possible values consists of an entire interval on a number line. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Random Variables Represents a possible numerical value from a random event Random Variables Discrete Random Variable Continuous Random Variable Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Continuous 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, The graph of f is the density curve. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Probability Density Function For f (x) to be a pdf f (x) > 0 for all values of x. The area of the region between the graph of f and the x – axis is equal to 1. Area = 1 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Probability Density Function is given by the area of the shaded region. a b Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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, Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 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.

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, and for any numbers a and b with a < b, Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 exists, Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 , is defined by Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 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.

of Continuous Random Variables Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 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 The standard deviation is Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Short-cut Formula for Variance Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Covariance between two discrete random variables: σxy =  [x – E(x)] [y – E(y)] f(xy) where: xi = possible values of the x discrete random variable yj = possible values of the y discrete random variable P(xi ,yj) = joint probability of the values of xi and yj occurring Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Let X and Y be random variables, The Covariance of X and Y is Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Correlation Coefficient The Correlation Coefficient shows the strength of the linear association between two variables where: ρ = correlation coefficient (“rho”) σxy = covariance between x and y σx = standard deviation of variable x σy = standard deviation of variable y Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Interpreting the Correlation Coefficient The Correlation Coefficient always falls between -1 and +1  = 0 x and y are not linearly related. The farther  is from zero, the stronger the linear relationship:  = +1 x and y have a perfect positive linear relationship  = -1 x and y have a perfect negative linear relationship Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. 4.3 Mean and Variance of Linear Combinations of Random Variables Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Expected Value of Linear Combinations This leads to the following: For any constant a, 2. For any constant b, Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Expected Value of Also Finally Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. If X and Y are independent, then Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Expected value of the sum of two discrete random variables: E(x + y) = E(x) + E(y) =  x P(x) +  y P(y) (The expected value of the sum of two random variables is the sum of the two expected values) Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Rules of Variance and This leads to the following: Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. If X and Y are random variables, then If X and Y are Independent random variables, then Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. If X and Y are Independent random variables, then If X1, X2, … Xn are Independent random variables, then Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Special Continuous Probability Densites Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

The Normal Distribution Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Standard Normal Distributions The normal distribution with parameter values is called a standard normal distribution. The random variable is denoted by Z. The pdf is The cdf is Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Standard Normal Cumulative Areas Standard normal curve 0 z Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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. Area to the left of 0.85 = 0.8023 b. P(Z > 1.32) Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Find the area to the left of 1.78 then subtract the area to the left of –2.1. = 0.9625 – 0.0179 = 0.9446 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Notation will denote the value on the measurement axis for which the area under the z curve lies to the right of Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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.

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 has a standard normal distribution. Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Let X be a normal random variable with = 0.2266 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 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.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. = 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 for normal Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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. = 0.9394 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

The Gamma Distribution Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc. The Gamma Function the gamma function Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 where the parameters satisfy The standard gamma distribution has Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 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 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.

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 chi-squared distribution with parameter v if the pdf of X is the gamma density with The pdf is Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 symbol is often used in place of “chi-squared.” Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

Other Continuous Distributions Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 where the parameters satisfy Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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, if the pdf of X is Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.

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 Copyright (c) 2004 Brooks/Cole, a division of Thomson Learning, Inc.