The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane Probability and Stochastic Processes Yates and Goodman Chapter 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  A c – complement  A-B difference  Mutually exclusive  Collectively exhaustive  DeMorgan’s Thm: (A  B) c = A c  B c

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] ≥ 0 P[S] = 1 P[A 1  A 2  …] = P[A 1 ] + P[A 2 ] … (for mutually exclusive events) P[B] =  P[{s i }]  Equally likely outcomes P[s i ] = 1/n (n possible states, s i )

The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane 1.4 Theorems of Probability  P[A  B] = P[A] + P[B] – P[A  B]  If A  B the P[A]  P[B]  For any event A, and event space {B 1,B 2,…B m }, P[A] =  P[A  B i ]

The Erik Jonsson School of Engineering and Computer Science © Duncan L. MacFarlane 1.5 Conditional Probability  Conditional Probability P[A|B] = P[A  B]/P[B]  Law of Total Probability P[A] =  P[A|B i ]P[B i ]  Bayes Thm P[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 events P[A  B] = 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 E i … E k – E i has n i outcomes – E has  k n i outcomes  Choose with replacement – n distinguishable objects – n k 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[S k,n ] = ( n k ) p k (1-p) n-k  Multiple outcomes P[N 1 =n 1,N 2 =n 2 …N r =n r ]=M  r p i n i where M= n!/n 1 !n 2 !...n r !  Reliability – Series – Parallel