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# Chapt4 - Chapter 4 Stat 211 Prof Parzen STANDARD...

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Chapter 4 Stat 211 Prof Parzen STANDARD DISTRIBUTIONS FOR APPLIED STATISTICS In statistical practice there are a small number of distinguished distributions which researchers use as models for observed data. The continuous distributions that are fundamental to modeling data are described using parameters and standard distributions such as Uniform (0,1), Exponential(1) and Normal(0,1). A continuous random variable U is said to have a (standard) Uniform(0,1) distribution if it has probability density ( ( ; 1, 0 1 0, otherwise; f y f y U y = = < < = We often use U as a symbol for a Uniform(0,1) random variable. We write U=Uniform(0,1) to mean the random variable U on left of the equation equals in distribution (has the same distribution) as the distribution on the right side of the equation. DEFINITION: Uniform(a,b) distributions where a<b are specified numbers called parameters of the distribution is defined ( ( ( ; , 1 , f y Uniform a b b a a y b = - < < . A continuous random variable Y is said to have (standard) Exponential(1) distribution if it has probability density ( ( ; , 0 0, otherwise. y f y f y Y e y - = = = We often use W as a symbol for an Exponential(1) random variable and write W =Exponential(1) DEFINITION: A random variable Y is said to have Exponential distribution with location parameter μ and scale parameter σ if Y W μ σ = + , where W = Exponential(1). Equivalently the probability density of Y is ( 29 ( 29 1 ; ; 1 exp , 0 , otherwise ; | , y f y Y f W y y f y Y μ σ σ μ μ σ σ μ σ - = - = - = = To emphasize that the probability density depends on parameters μ and σ we write it ( 29 1 ; | , ; . y f y Y f W μ μ σ σ σ - =

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Equivalent notation: ( 1 Y Exponential μ σ = + Very important in practice is case 0 μ = . Then equivalent statements are: Y σ = Exponential (where = means equal in distribution), Y is Exponential ( 0, σ . STANDARD NORMAL DISTRIBUTION A continuous random variable Y is said to have (standard) Normal(0,1) distribution if it has probability density ( ( ( ; f y f y Y y ϕ = = where ( y ϕ has a universal meaning as ( 29 ( 29 2 1 exp .5 2 y y ϕ π = -
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Chapt4 - Chapter 4 Stat 211 Prof Parzen STANDARD...

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