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Lecture3_412 - AMS412 Lecture Notes#3 Review of...

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1 AMS412’ Lecture’Notes’ #3’ Review of Probability (continued) (3) Normal Distribution Q. Who invented the normal distribution? * Left: Abraham de Moivre ’ (26’May’1667’in’ Vitry-le-François ,’ Champagne ,’ France ’ –’ 27’November’1754’in’ London ,’ England )’ *From’the’Wikipedia’ * Right: Johann Carl Friedrich Gauss ’ (30’April’1777’ –’ 23’February’1855) ’ <i> Probability Density Function (p.d.f.) ) , ( ~ 2 σ μ N X :’ X’follows’normal’distribution’of’mean’ ’ and’variance’ 2
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2 2 2 2 ) ( 2 1 ) ( σ μ π = x e x f ,’ R x x , = b a dx x f b X a P ) ( ) ( =’area’under’the’pdf’curve’bounded’by’a’and’b <ii> Cumulative Distribution Function (c. d .f.) = = x dt t f x X P x F ) ( ) ( ) ( ) ( )]' ( [ ) ( x F d x F x f = = (4). Mathematical Expectation (Review). Continuous’random’variable:’ [( ) ] ()() EgX gx f xdx −∞ = Discrete’random’variable:’ ] ()( ) all gxPX x ==
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3 Properties of Expectations: (1) E(c)’=’c,’where’c’is’a’constant (2) E[c*g(X)]’=’c*E[g(X)],’where’c’is’a’constant (3) E[g(X)+h(Y))]=’E[g(X)]+E[h( Y)],’for’ any ’ X&Y’ (4) E[g(X)*h(Y)]’=’E[g(X)]*E[h(Y)] ,’ if’X’&’Y’are’ independent ’ –otherwise’it’is’usually’ not’true. ’ Special case: 1) (population) Mean :’ () ( ) EX x f xd x μ −∞ == Note: E(aX+b) =aE(X)+b, where a & b are constants 2) (population) Variance :’ ’ Var(X)’=’ 22 2 2 2 [( ) ] ( ) ( ) ( ) [ ( )] x fxd x EX σ −∞ = = = Note: Var(aX+b) = a 2 Var(X), where a & b are constants 3) Moment generating function: = = dx x f e e E t M tx tX X ) ( ) ( ) ( ,’ when’X’is’continuous .’ = ࠵± = ࠵± ࠵±± ࠵±²²³´µ¶ ࠵±²³´µ ࠵± For’normal’distribution,’ ), , ( ~ 2 N X 2 2 2 ) ( 2 1 ) ( π = x e x f ,’ < < x 2 2 2 1 ) ( ) ( t t tx X e x f e t M + = =
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