The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane Probability and...
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Transcript of The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane Probability and...
The Erik Jonsson School of Engineering and Computer Science
© Duncan L. MacFarlane
Probability and Stochastic ProcessesYates and GoodmanChapter 1 Summary
Duncan MacFarlane
The Erik Jonsson School of Engineering and Computer Science
© Duncan L. MacFarlane
1.1 Set Theory
Venn Diagrams -- subset of -- element of -- union -- intersection Ac – complement
A-B difference Mutually
exclusive Collectively
exhaustive DeMorgan’s Thm:
(AB)c = AcBc
The Erik Jonsson School of Engineering and Computer Science
© Duncan L. MacFarlane
1.2 Applying Set Theory to Probability
Experiments Outcomes Sample Space (S)
– Finest grain
Events Event Space
The Erik Jonsson School of Engineering and Computer Science
© Duncan L. MacFarlane
1.3 Probability
Axioms:P[A] ≥ 0P[S] = 1P[A1A2…] = P[A1] + P[A2] … (for mutually exclusive events)
P[B] = P[{si}]
Equally likely outcomesP[si] = 1/n (n possible states, si)
The Erik Jonsson School of Engineering and Computer Science
© Duncan L. MacFarlane
1.4 Theorems of Probability
P[AB] = P[A] + P[B] – P[AB] If AB the P[A] P[B] For any event A,
and event space {B1,B2,…Bm},
P[A] = P[ABi]
The Erik Jonsson School of Engineering and Computer Science
© Duncan L. MacFarlane
1.5 Conditional Probability
Conditional ProbabilityP[A|B] = P[AB]/P[B]
Law of Total ProbabilityP[A] = P[A|Bi]P[Bi]
Bayes ThmP[B|A] = P[A|B]P[B]/P[A]
The Erik Jonsson School of Engineering and Computer Science
© Duncan L. MacFarlane
1.6 Independent Events
Definition of independent eventsP[AB] = P[A]P[B]
– Independence is not mutually exclusive
– Extensions to more than 2 events
1.7 tree diagrams
The Erik Jonsson School of Engineering and Computer Science
© Duncan L. MacFarlane
1.8 Counting Methods
Fundamental Principle of Counting– Experiment E
– Sub-Experiments Ei … Ek
– Ei has ni outcomes
– E has k ni outcomes
Choose with replacement– n distinguishable objects– nk ways to choose (with replacement) a
sample of k objects
The Erik Jonsson School of Engineering and Computer Science
© Duncan L. MacFarlane
1.8 Counting Methods: Permutations and Combinations
k-permutations … order matters!(n)k = (n)(n-1)(n-2) … (n-k+1)
= n!/(n-k)! k-combinations … order doesn’t
matter!(n
k) = (n)k/k! = n!/n!(n-k)!– “n choose k”
The Erik Jonsson School of Engineering and Computer Science
© Duncan L. MacFarlane
1.9 Independent Trials
Probability of k successes out of n trials P[Sk,n] = (n
k) pk(1-p)n-k
Multiple outcomesP[N1=n1,N2=n2…Nr=nr]=M rpi
ni
where M= n!/n1!n2!...nr! Reliability
– Series– Parallel