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Unformatted text preview: 1 Continuous Probability Distributions k MGMT 670: Business Analytics Krannert School of Management Purdue University 2 Continuous Random Variable Assumes any value in an interval or a collection of intervals on the real line e.g., stock returns, lifetime of a part, time between arrivals Is it possible to define the probability of the continuous random variable taking a particular value? The probability for continuous random variables is defined for ___________. 3 Continuous Probability Density Function Informally, a smoothed histogram, showing relative frequencies, f ( x ), of all values of the random variable, X . However, f(x) is not probability . Properties of the probability density function ( ) 1, ( ) f x dx f x + = f ( x ) a b x (Value, pdf ) Value pdf 4 Probability of Continuous Random Variable Probability is the area under curve! 19841994 T/Maker Co. f ( x ) a b x x 1 x 2 = 2 1 ) ( ) ( 2 1 x x dx x f x X x P Properties of the probability density function ( ) 1, ( ) f x dx f x + = 5 Normal Probability Distribution Describes many continuous random processes or phenomena. Can be used to approximate some discrete probability distributions such as Binomial and Poisson distributions. Basis for classical statistical inference. 6 Normal Probability Density Function Normal Probability Density Function (PDF) where: x = value of random variable ( x ) = mean = standard deviation = 3.14159 e = 2.71828 2 2 1 2 1 ) (  = x e x f 7 Characteristics of Normal Probability Distribution The Normal Probability Density Function (PDF) Bellshaped and symmetric Mean, median, and mode are equal. Middle spread is 1.35 . Infinite Range 8 Some Common Intervals of Normal Probability Distribution x +  2 3 + 2 + 3 68% 95% 99.7% 9 Parameters of Normal Probability Distribution Mean ( ) and standard deviation ( ) determine the location and shape of the distribution. x f ( x ) C A B For a normal population, if we know the mean and standard deviation, we have the complete information about the population. 10 Normal Probability Probability is the area under curve. c d x f(x) = d c dx x f d X c P ) ( ) ( 11 Standard Normal Probability Distribution Normal distribution with a mean of zero and a standard deviation of one. z is commonly used to designate the standardized normal random variable. Conversion rule Measures the number of standard deviations the value x is from . x z  = 12 Standardization Example P(5 X 17.5) N(5,102) 13 Using Table to Find Normal Probabilities Standardize, then use standard normal table....
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 Spring '11
 Tawarmalani
 Management

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