M670_5(Continuous Distributions)

# M670_5(Continuous Distributions) - 1 Continuous Probability...

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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! © 1984-1994 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 ) μ = mean σ = standard deviation π = 3.14159 e = 2.71828 2 2 1 2 1 ) ( -- = σ μ π σ x e x f 7 Characteristics of Normal Probability Distribution The Normal Probability Density Function (PDF) • Bell-shaped and symmetric • Mean, median, and mode are equal. • Middle spread is 1.35 . • Infinite Range 8 Some Common Intervals of Normal Probability Distribution x μ μ+ σ μ-σ μ-2 σ μ-3 σ μ+ 2 σ μ+ 3 σ 68% 95% 99.7% 9 Parameters of Normal Probability Distribution § Mean ( µ ) and standard deviation ( σ ) 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 μ σ- = 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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M670_5(Continuous Distributions) - 1 Continuous Probability...

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