handout0 - ISyE8843A, Brani Vidakovic 1 1.1 Handout 0 Some...

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ISyE8843A, Brani Vidakovic Handout 0 1 Some Important Continuous Distributions 1.1 Uniform U ( a,b ) Distribution Random variable X has uniform U ( a,b ) distribution if its density is given by f ( x | a,b ) = 1 b - a , a x b 0 , else F ( x | a,b ) = 0 , x < a x - a b - a , a x b 1 , x > b Moments: EX k = 1 b - a b k +1 - a k +1 k +1 , k = 1 , 2 ,... Variance V arX = ( b - a ) 2 12 . Characteristic Function ϕ ( t ) = 1 b - a e itb - e ita it . If X ∼ U ( a,b ) then Y = X - a b - a ∼ U (0 , 1) . Typical model: Rounding (to the nearest integer) Error is often modeled as U ( - 1 / 2 , 1 / 2) If X F , where F is a continuous cdf, then Y = F ( X ) ∼ U (0 , 1) . 1.2 Exponential E ( λ ) Distribution Random variable X has exponential E ( λ ) distribution if its density and cdf are given by f ( x | λ ) = λe - λx , x 0 0 , else , F ( x | λ ) = 1 - e - λx , x 0 0 , else Moments: EX k = k ! λ k , k = 1 , 2 ,... Variance V arX = 1 λ 2 . Characteristic Function ϕ ( t ) = λ λ - it . Exponential random variable X possesses memoryless property P ( X > t + s | X > s ) = P ( X > t ) . Typical model: Lifetime in reliability. Alternative parametrization, λ 0 as scale. f ( x | λ 0 ) = 1 λ 0 e - x λ 0 , EX k = k !( λ 0 ) k , k = 1 , 2 ,... , V arX = ( λ 0 ) 2 . 1.3 Double Exponential DE ( μ,σ ) Distribution Random variable X has double exponential DE ( μ,σ ) distribution if its density and cdf are given by f ( x | μ,σ ) = 1 2 σ e -| x - μ | , F ( x | μ,σ ) = 1 2 1 + sgn ( x - μ )(1 - e -| x - μ | · , x,μ R ; σ > 0 . Moments: EX = μ , EX 2 k = [ μ 2 k (2 k )! + μ 2( k - 1) σ 2 (2( k - 1))! + ... σ 2 k 1 ](2 k )! EX 2 k +1 = μ 2 k +1 (2 k + 1)! Variance V arX = 2 σ 2 . Characteristic Function ϕ ( t ) = e iμt 1+( σt ) 2 . 1
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1.4 Normal (Gaussian) N ( μ,σ 2 ) Distribution Random variable X has normal N ( μ,σ 2 ) distribution with parameters μ R (mean, center) and σ 2 > 0 (variance) if its density is given by f ( x | α,β ) = 1 2 πσ 2 e - ( x - μ ) 2 2 σ 2 , - ∞ < x < Moments: EX = μ,E ( X - μ ) 2 k - 1 = 0; E ( X - μ ) 2 k = (2 k - 1)(2 k - 3) ... 5 · 3 · 1 · σ 2 k = (2 k - 1)!! σ 2 k , k = 1 , 2 ,... Characteristic function ϕ ( t ) = e iμt - t 2 σ 2 / 2 . Z = X - μ σ has standard normal distribution φ ( x ) = 1 2 π e - x 2 2 . The cdf of standard normal distribution is a special function Φ( x ) = R x -∞ φ ( t ) dt and its values are tabulated in many introductory statistical texts. Standard half-Normal distribution is given by f ( x ) = 2 φ ( x ) 1 ( x 0) . 1.5 Chi-Square χ 2 n Distribution Random variable X has chi-square χ 2 n distribution distribution with n degrees of freedom if its density is given by f ( x ) = ( 1 2 n/ 2 Γ( n/ 2) x n/ 2 - 1 e - x/ 2 , x > 0 0 , else Moments: EX k = n ( n + 2) ... ( n + 2( k - 1)) ,k = 1 , 2 ,.... Expectation
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handout0 - ISyE8843A, Brani Vidakovic 1 1.1 Handout 0 Some...

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