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COMPLETE Statistics Notes - Part 5 (got 4.0 in the course)

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ECON 1151
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Final—Discrete & Continuous Random Variables Keely Henesey Continuous Random Variables Discrete Random Variables b Expected Value ∑ xP(x) E X ]=∫af (x)dx x b 2 E (X−μ) 2= ∫x−μ) f (x)dx ∑ (x−μ) P(x)= ∑ (x−μ)xP(x) a x x Variance b 2 2 E [X2]−μ = ∫ f (x)dx−μ 2 ∑ x P x – μ a x Standard Deviation σ= σ√ 2 b Expected Value of Functions  E [(X)=]g∫x)f (x)dx ∑ g(x)P(x) of Random Variables a x μ =E [a+bX ]=a+bμ Y X 2 2 2 Let  Y=a+bX ] σ YVar a+bX =b σ X 2 σ Y ∣b∣σX= √ Y Let  b=0 ]  in  E a =a W=a+bX ] Var(a)=0 Thus:[W=a ] Let  a=0 ]  in  E bX ]=bμ X W=a+bX ] 2 2 Thus:[W=bX ] Var bX =b σ x Let  [=−μ /X X]  and  X−μ μ [=1/σ ]  in  E X = X − 1 μX=0 X [ ]σX σX σX Z=a+bX ] X−μ X−μ X 1 2 Thus: Z= X Var( )σ = σ 2σ X1 [ σ X ] X X Jointly Distributed Discrete Random Variables P x, y) P(y∣x)= P(x) Conditional Probability  Distribution P(x∣y)= P(x, y) P(y) Final—Discrete & Continuous Random Variables Keely Henesey μY∨X=E [Y∨X ]=∑ (y∨x)P(y ∣x) Conditional Mean y 2 2 2 Conditional Variance σY∨X =E [(μ Y∨X )∨X =] ∑ ((y−μ Y∨X)∣x)P(y ∣x) y Independent If… P(y∣x)=P(y) P(x∣y)=P(x) Jointly Distr
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