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
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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
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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!
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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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