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# IEN 310 Test 1 IEN 310

UM

GPA 3.7

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This 7 page Test Prep (MCAT, SAT...) was uploaded by Connor McCullough on Friday February 13, 2015. The Test Prep (MCAT, SAT...) belongs to IEN 310 at University of Miami taught by Professor Sharit in Spring2015. Since its upload, it has received 109 views. For similar materials see Introduction to Probability in Industrial Engineering at University of Miami.

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Date Created: 02/13/15

Chapter 1 Set Theory x is element in A x E A A is subset of B A c B Null set 0 Compliment of A AC Union of A and B A u B Intersection A m B Difference AB Mutually Exclusive Ai Aj Q Collectively Exhaustive A1 uA2uuAnu Boolean Algebra Commutative laws AUBBUA A m B B m A Associative laws AUBUCAUBUC A B CA B C Distributive laws AuB CAuB AuC A BuCA BuA C Idempotent laws AuAA A AA Absorption laws AuA BA A A BA B 1A 2AU A 3800 4A BCACuBC DeMorgan s Law AuBCAC BC Set TheorvProbabiitv outcome any observation of the experiment sample space 8 all possible outcomes event set of related outcomes from experiment Axioms of Probability Axiom 1 For any event A PA 2 O Axiom 2 PS 1 Axiom 3 For any countable collection A1 A2 of mutually exclusive events PA1uA2uPA1PA2 Theorem 12 For mutually exclusive events A1 and A2 114 1113 11111 1113 Theorem 13 lfA A1UA2U39 LJAm and A 1 A ofor i j then 1 PM I llzl ll Theorem 14 The probability measure P satisfies a PQ O b PAC 1 PA c For any A and B not necessarily mutually exclusive PAUB PAPB PA B d If A c B then PA s PB Theorem 15 The probability of an event B s1 s2 sm is the sum of the probabilities of the outcomes contained in the event 21 PM 2 PM HI Theorem 16 For an experiment with sample space S s1 sn where each outcome si is equally likely I I M igln 1 lt5 139 lt1 12 Conditional Probability Definition 15 The conditional probability of the event A given the occurrence of the event B is 1 1 Fl 13 111113 g 1 b Theorem 17 A conditional probability measure PAIB has the following properties that correspond to the axioms of probability Axiom 1 PAIB 2 O Axiom 2 PBIB 1 Axiom3lfAA1uA2uwithAi Ajofori6j then PAIB PA1IB PA2IB Pa Mons Theorem 18 For a partition B B1 B2 and any event A in the sample space let Ci A 1 Bi For i 6 j the events Ci and C are mutually exclusive and A C1 u C2 u Theorem 19 For any event A and partition B1 B2 Bm PA i1 PA 1 Bi Theorem 110 The Law of Total Probability For a partition B1 B2 Bm with PBi gt O for all i and for some event A in the sample space 2 1 PM 2 11111341111 Theorem 111 Bayes Theorem 1 l H B 1 M13 PM For partitioned B rim ll 3 1l 3 l 2 1 Al 1321132J Independence Events A and B are independent if and only if PAB PAPB More than Two Independent Events If n 2 3 the events A1 A2 An are independent if and only if a All collections of n 1 events chosen from A1 A2 An are mutually independent b PA1 A2 39 39 39 H An PA1PA2 39 39 39 PAn Note If the events A1 A2 An are not independent then PA1 n A2 n n An PA1PA2 A1PA3A1 n A2 PAnIA1 n A2 n Am Chapter 2 Theorem 21 Fundamental Principle of Counting lf subexperiment A has n possible outcomes and subexperiment B has k possible outcomes then there are nk possible outcomes when you perform both subexperiments More generally if an experiment Ehas ksubexperiments E1 Ek where E has ni outcomes then E has l39lk i1ni outcomes Theorem 22 The number of kpermutations of n distinguishable objects is K 11 l I I 1 39l I II rll From this expression it should be clear that when k n then n nn n II 1 k permutations of n objects n choose k binomial coefficient Theorem 23 and Definition 21 39 n II FEST ITTA ll 112 lI II A nilH rzzvru A II A39 Theorem 24 Sampling with Replacement Given m distinguishable objects there are mn ways to choose with replacement an ordered sample of n objects Note that this result can be derived from the fundamental principle of counting Theorem 25 For n repetitions of a subexperiment with sample space S so sm1 there are mn possible observation sequences ie outcomes of the sequential experiment Theorem 26 The number of observation sequences for n subexperiments with sample space S 0 1 with O appearing no times and 1 appearing n1 n no times is II 111 Theorem 27 For n repetitions of a subexperiment with sample space Ssub sO sm 1 the number of length n no nm 1 observation sequences with si appearing ni times is II Ill II lIn 1 lluillllulln 1 Definition 22 For Multinomial Coefficient 39 l 221 1 II II IIIsziL l 11 HuiHimjlm 1 2 II II 1 hiflzrl39u39isr Independent Trials Theorem 28 The probability of no failures and n1 successes in n no n1 trials is II II w an n n w um 21 ll 1111 211 ll 1l 1 l ll 1 1 I I Theorem 29 A subexperiment has sample space Ssub so sm1 With Psi pi For n 110 nm1 independent trials the probability of 111 occurrences of si i 0 m 1 is II N j nm 1 Idllglwullm 1 1 NJ 1 ILLUMIIIH 1 1 Reliability Analysis Probability operation succeeds for components in series I lll39l 1 u391ugu39n 1 Probability operation succeeds for components in parallel I lll39 l 1 lll l 1 l Chapter 3 Discrete Random Variable X is a discrete random variable if the range of X is a countable set SX X1 X2 Set of possible values can be listed even if list is infinitely long Finite Random Variable Range of values is a finite set Probability Mass Function of discrete random variable X PXX PX X Theorem 31 For a discrete random variable X With PMF PXX and range SX a For any X PXX 2 O E 11 l b c For any event B C SX the probability that X is in the set B is 1113 2 I thm MB Families of Discrete Random Variables Bernoulli p Random Variable l 1 I I XU39 I r l U ifH 139 HM Geometric p Random Variable R Hill p quot 1I 12 1 VLF 39 UniH I39u lm Binomial np Random Variable f quot l39 quot 39 ll 1 X 1394 1 l 1 I Pascal kp Random Variable I l I l klill 1quot 11y f r j Discrete Uniform kl Random Variable 4 139quotI39 11quot 1139139 1 H131 39 39 Lotu ru zm Poisson 0 Random Variable Ur H I l l 3 1 3 quot l ufn ru zm Cumulative Distribution Function CDF FXX PX s X Theorem 32 For any discrete random variable X with range SX X1 X2 satisfying X1 5 X2 5 a FX Oo O and FXOO 1 i e going from left to right on the XaXis FXX starts at zero and ends at one b For all X 0 z X FXX O z FXX i e the CDF never decreases as it goes from left to right c For Xi E SX and an arbitrarily small positive number FXXi FXXi E PXXi i e for a discrete random variable X there is a jump at each value of Xi E SK and the height of this jump at Xi is PXXi d FXX FXXi for all X such that Xi s X lt Xi1 i e between jumps the graph of the CDF of the discrete random variable X is a horizontal line Theorem 33 For all b 2 a FXb FXa Pa lt X s b Difference between the CDF evaluated at two points is the probability that the random variable takes on a value between these two points Averages and EXpected Value Definition 313 EXpected Value EX IX Z I39I A39IfJf l I Dquot EXpected Values For Families Bernoulli EX p Geometric EX lp Binomial EX np Pascal EX Xp Discrete uniform EX kl2 Poisson EX 0t Functions of a Random Variable Derived Random Variable Y gX y 39 zi Z 131ij PMF I Iy39 I I39g Expected Value of Derived Random Variable El39 1y Z 4Vr1quotrf l I Dv y EX 1 39 EIX b MELV I Variance and Standard Deviation Variance VarX E X AXY Standard Deviation oX sqrtVarX Computational Value of Variance VarX EXZ sz EXZ EX2 Variances By Family Bernoulli VarX pl p Geometric VarX l pp2 Binomial VarX npl p Pascal VarX kl pp2 Discrete Uniform VarX l kl k2 12 Poisson VarX O

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