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### myppch04

Course: STAT 344, Summer 2008
School: George Mason
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Word Count: 1466

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4 Continuous Chapter Random Variables and Probability Distributions 4.1 Continuous Random Variables and Probability Distributions Continuous Random Variables A random variable X is continuous if its set of possible values is an entire interval of numbers (If A &lt; B, then any number x between A and B is possible). Probability Distribution Let X be a continuous rv. Then a probability distribution or...

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4 Continuous Chapter Random Variables and Probability Distributions 4.1 Continuous Random Variables and Probability Distributions 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). 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 c X b ) = c f ( x)dx a b The graph of f is the density curve. 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 Probability Density Function P (a c X b) is given by the area of the shaded region. y = f ( x) a b 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 Ac x B f ( x; A, B ) = B - A 0 otherwise c c 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 = P(a b) X < b) = P ( a < X < b) 4.2 Cumulative Distribution Functions and Expected Values 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) = x - f ( y )dy For each x, F(x) is the area under the density curve to the left of x. 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 c X b ) = F (b) - F (a ) 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 c x) exists, F c x) = f ( x). ( 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 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 . Expected Value The expected or mean value of a continuous rv X with pdf f (x) is X = E ( X ) = - x f ( x)dx 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 Variance and Standard Deviation The variance of continuous rv X with pdf f(x) and mean is 2 X = V ( x) = - (x - ) 2 2 f ( x)dx = E[( X - ) ] The standard deviation is X = V ( x). Short-cut Formula for Variance V (X ) = E X ( ) - [ E ( X )] 2 2 4.3 The Normal Distribution Normal Distributions A continuous rv X is said to have a normal distribution with parameters and , where - < < 0 < , if the pdf of X is 1 - ( x - )2 /(2 2 ) f ( x) = e 2 and < x <2 - 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 - Standard Normal Cumulative Areas Standard normal curve Shaded area = (z ) 0 z Standard Normal Distribution Let Z be the standard normal variable. Find (from table) a. P ( Z c 0.85) Area to the left of 0.85 = 0.8023 b. P(Z > 1.32) 1 - P ( Z 1.32) = 0.0934 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 n 1.78) - P ( Z = 0.9625 0.0179 = 0.9446 -2.1) z Notation measurement axis for which the area under the z curve lies to the right of z . Shaded area = P ( Z z ) = z will denote the value on the 0 z 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 ) ] = 2P(z < Z ) 1 = 0.8132 P(z < Z ) = 0.9066 z = 1.32 Nonstandard Normal Distributions If X has a normal distribution with mean and standard deviation , then X - Z= has a standard normal distribution. Normal percentage Curve Approximate of area within given standard deviations (empirical rule). 99.7% 95% 68% Ex. Let X be a normal random variable with = 80 and = 20. Find P ( X c 65). P( X 65 ) = P Z 65 - 80 20 = P( Z = 0.2266 -.75 ) 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. P ( 3.75 h X 3.75 -6 9 ) = P 1.5 Z 9-6 1.5 = P ( -1.5 Z 2) = 0.9772 0.0668 = 0.9104 Percentiles of an Arbitrary Normal Distribution (100 (100p)th percentile p )th for =+ , ) for normal ( standard normal 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 . P( X x) + 0.5 - np x npq 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 most 375 students pass. = np = 500(.72) = 360 = npq = 500(.72)(.28) 10 P ( X ~ 375) 375.5 - 360 = (1.55) 10 = 0.9394 4.4 The Gamma Distribution and Its Relatives The Gamma Function For > 0, the gamma function ( ) is defined by ( ) = x 0 -1 - x e dx Gamma Distribution A continuous rv X has a gamma distribution if the pdf is 1 f ( x; , ) = ( ) 0 x -1 - x / e x 0 otherwise where the parameters satisfy > 0, > 0. The standard gamma distribution has = 1. 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 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; ) = y -1 - y 0 e dy ( ) Exponential Distribution A continuous rv X has an exponential distribution with parameter if the pdf is f ( x; ) = e - x x 0 0 otherwise Mean and Variance The mean and variance of a random variable X having the exponential distribution 1 = = = = 2 2 1 2 Probabilities from the Gamma Distribution Let X have an exponential distribution Then the cdf of X is given by F ( x; ) = 0 1- e - x x<0 x 0 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 = . 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 f ( x; v) = 2v / 2 (v / 2) 0 x ( v / 2) -1 - x / 2 e x 0 x<0 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." 4.6 Probability Plots 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. Probability Plot [100(i - .5) / n]th percentile ith smallest sample of observation the distribution , If the sample percentiles are close to the corresponding population distribution percentiles, the first number will roughly equal the second. 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 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
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