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SIE 305 Notesheet Final

SIE 305 Notesheet Final - Chapter 1 Introduction Stuff...

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Chapter 1 Introduction Stuff Mutually Exclusive – Two events A 1 and A 2 are mutually exclusive if and only if A 1 ∩ A 2 = Ø. Events A 1 , A 2 , A 3 ,… are mutually exclusive if and only if A i ∩ A j = Ø for i ≠ j. ( 29 ( 29 ! ! ! ! ! r n r n r n C r n n P r n r n - = = - = Permutations of Indistinguishable Objects ! ! ! ! 2 1 k n n n n Chapter 2 Probability Laws General Addition Rule [ ] [ ] [ ] [ ] ° 2 1 2 1 2 1 A A P A P A P A A P + + = Conditional Probability [ ] [ ] [ ] 1 2 1 1 2 | A P A A P A A P = Independent Events (events A 1 and A 2 are independent if and only if the below condition is met) [ ] [ ] [ ] 2 1 2 1 A P A P A A P = [ ] [ ] [ ] [ ] [ ] [ ] 0 if | and 0 if | 2 1 2 1 1 2 1 2 = = A P A P A A P A P A P A A P Bayes’ Theorem [ ] [ ] [ ] [ ] [ ] = = n i j j j j j A P A B P A P A B P B A P 1 | | | Events A i are mutually exclusive. P[B] is not zero. Chapter 2 Probability Laws Convergence of Geometric Series 1 1 ) 1 ( terms, first For the 1 | | provided 1 to converges series The series geometric a be Let 1 1 1 1 - - = < - = - = - r r r a ar n r r a ar n n k k k k Chapter 3 Discrete Distribution Expected Value, Var, Std Dev [ ] ( 29 [ ] [ ] [ ] ( 29 X X E X E X E X x xf X E Var Deviation Standard Var ) ( 2 2 2 2 2 x all = - = - = = = = σ σ μ σ μ Moment Generating Function [ ] [ ] k t k X k t X X E dt t m d X e E t m = = = 0 ) ( ) ( Chapter 4 - = σ μ x z For exponential distribution Beta = 1/Lamda Binomial fixed n trials p is probability of success x denotes successes obtained in n trials Negative Binomial trials observed until r successes (r is fixed) x denotes number of trials needed for r successes Hypergeometric random sample size n, from collection of N r have traits of interest (N-r do not) x is the number of objects in n that have trait Geometric Properties series of trials with success of fail outcomes (Bernoulli) trials are independent and probability of success p remains constant x denotes trials necessary until 1 st success Transformation of Variables ) ( ), ( ) ( )) ( ( ) ( 1 1 1 y g x x g y dy y dg y g f y f x y - - - = = = Chapter 5 Necessary Conditions for Discrete Joint Density - ∑∑ = x all y all 1 ) , ( 0 ) , ( y x f y x f XY XY Discrete Marginal Densities = = x all y all ) , ( ) ( ) , ( ) ( y x f y f y x f x f XY Y XY X Continuous Joint Density and Continuous Marginal Density (Change summations to integrals and set limits as –infinity to infinity).
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Independence [ ] [ ] [ ] ) ( ) ( ) , ( 2 1 2 1 y f x f y x f A P A P A A P Y X XY = = Expected Value [ ] [ ] [ ] - - - - - - = = = dydx y x yf Y E dydx y x xf X E dydx y x f y x H y x H E XY XY XY ) , ( ) , ( ) , ( ) , ( ) , ( Covariance [ ] [ ] [ ] [ ] [ ] [ ] [ ] 1 1 - ) )( ( ) , ( the t, independen are Y and X If ) )( ( ) , ( = = - = - - = XY XY Y X VarY VarX y x Cov Y E X E XY E Y E X E XY E y x E y x Cov ρ ρ μ μ Conditional Density 0 ) ( ) ( ) , ( 0 ) ( ) ( ) , ( | | = = x f x f y x f f y f y f y x f f x x XY x Y y y XY y X Curve of regression Mean value of X given Y=y, denoted by M X|y is called the curve of regression.
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