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