# Week3 - Stat231 William Marshall Stat231 William Marshall...

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Stat231 William Marshall Stat231 William Marshall May 16, 2010

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Stat231 William Marshall Week 3 Goals: Review continuous probability models (STAT230) Introduce the likelihood function and maximum likelihood estimation
Stat231 William Marshall Continuous random variables With continuous random variables, probabilities are dicussed in terms of intervals Probability density function R b a f ( y ) dy = P ( a Y b ) , a b S 0 f ( y ) y R y f ( y ) = 1

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Stat231 William Marshall Exponential distribution Describes arrival times of poisson process Memoryless property If Y Exp ( θ ) (0 < θ = λ - 1 ) f ( y ) = e - y θ θ , y > 0 E ( Y ) = θ sd ( Y ) = θ
Stat231 William Marshall Gaussian distribution Occurs in nature Useful for describing averages If Y G ( μ, σ ) ( σ > 0) f ( y ) = 1 σ 2 π e - ( y - μ ) 2 2 σ 2 , y R E ( Y ) = μ sd ( Y ) = σ

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Stat231 William Marshall Using tables Has numbers in the form P ( Z < z ) for positive values of z Use symmetry of Gaussian distribution Draw pictures!
Stat231 William Marshall Properties Suppose X 1 G ( μ 1 , σ 1 ) , X 2 G ( μ 2 , σ 2 ) If Y = aX 1 + b then Y G ( a μ 1 + b , | a | σ 1 ) If Y = X 1 + X

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Week3 - Stat231 William Marshall Stat231 William Marshall...

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