This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 1 Introduction Stuff Mutually Exclusive Two events A 1 and A 2 are mutually exclusive if and only if A 1 A 2 = . Events A 1 , A 2 , A 3 , are mutually exclusive if and only if A i A j = 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 = [ ] [ ] [ ] [ ] [ ] [ ] if  and 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 = = = = = = Moment Generating Function [ ] [ ] k t k X k t X X E dt t m d X e E t m = = = ) ( ) ( Chapter 4  = 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 (Nr 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 ) , ( ) , ( 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). 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...
View
Full
Document
This note was uploaded on 04/02/2008 for the course SIE 305 taught by Professor Leeming during the Spring '07 term at University of Arizona Tucson.
 Spring '07
 Leeming

Click to edit the document details