Stat 130 – Intro to Math Stat for CS Chapter 2 Reviewer
Probability Models ďƒ˜ Abstract Model - describing of the essential properties of a phenomenon that is formulated in mathematical terms ďƒ˜ Types: 1) Deterministic Model - describes a phenomenon through known relationships among states and events - will produce the same output given the same input - can’t describe outcomes in a game of chance - does not leave any room for RANDOM VARIATOIN 2) Probability/Stochastic Model - describes a phenomenon by assigning a likelihood of occurrence to the different possible outcomes of the process Basic Concepts of Probability ďƒ˜ Random Experiment – possible outcome are well-defined but the outcome is unpredictable ďƒ˜ Sample Space (Ί) - collection of all possible outcomes of a random experiment - an element is called a sample point (đ?œ”) ďƒ˜ Event - subset of the sample space whose probability is defined ďƒ˜ Impossible Event - đ?œ™ ďƒ˜ Sure Event - (Ί) ďƒ˜ Mutually exclusive events - đ??´ ∊ đ??ľ = đ?œ™ -> A and B have no elements in common (disjoint set) ďƒ˜ ________’s Axiomatic Definition of Probability 1. đ?‘‚ ≤ đ?‘ƒ(đ??´) ≤ 1 for any event A 2. đ?‘ƒ(Ί) = 1 3. For mutually exclusive events đ?‘ƒ(đ?‘˘đ?‘›đ?‘–đ?‘œđ?‘› đ?‘œđ?‘“ đ?‘Žđ?‘™đ?‘™ đ?‘’đ?‘Łđ?‘’đ?‘›đ?‘Ąđ?‘ ) = ∑ đ?‘ƒ(đ?‘œđ?‘“ đ?‘Žđ?‘™đ?‘™ đ?‘’đ?‘Łđ?‘’đ?‘›đ?‘Ąđ?‘ ) Approaches to Assigning Probabilities A. Classical Approach (Classical Probability or A Priori)
đ?‘ƒ(đ??´) =
# đ?‘œđ?‘“ đ?‘’đ?‘™đ?‘’đ?‘šđ?‘’đ?‘›đ?‘Ąđ?‘ đ?‘–đ?‘› đ??´ # đ?‘œđ?‘“ đ?‘’đ?‘™đ?‘’đ?‘šđ?‘’đ?‘›đ?‘Ąđ?‘ đ?‘–đ?‘› Ί
= portion of elements possessing the characteristic of interest
B. Relative Frequency Approach (A Posteriori) # đ?‘œđ?‘“ đ?‘Ąđ?‘–đ?‘šđ?‘’đ?‘ đ?‘’đ?‘Łđ?‘’đ?‘›đ?‘Ą đ??´ đ?‘œđ?‘?đ?‘?đ?‘˘đ?‘&#x;đ?‘’đ?‘‘ # đ?‘œđ?‘“ đ?‘Ąđ?‘–đ?‘šđ?‘’đ?‘ đ?‘’đ?‘Ľđ?‘?đ?‘’đ?‘&#x;đ?‘–đ?‘šđ?‘’đ?‘›đ?‘Ą đ?‘¤đ?‘Žđ?‘ đ?‘&#x;đ?‘’đ?‘?đ?‘’đ?‘Žđ?‘Ąđ?‘’đ?‘‘ C. Subjective Approach (Bayesian Approach) - using intuition, personal beliefs, other indirect info. đ?‘’đ?‘šđ?‘?đ?‘–đ?‘&#x;đ?‘–đ?‘?đ?‘Žđ?‘™ đ?‘ƒ(đ??´) = âˆŤ đ?‘›(đ??´) =
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Rules of Counting (Combinations) 1) Permutation: arrangement of ordering of all or part of a set of objects : # of permutations of n distinct objects taken r at a time is: đ?‘›!
đ?‘›đ?‘ƒđ?‘&#x; = (đ?‘›âˆ’đ?‘&#x;)! : # of distinct permutations of n things of which n1, are of are kind, n2 are of a second kind. . . nk of a kth kind is: đ?‘›! đ?‘›1 !đ?‘›2 ! ...đ?‘›đ?‘˜ !
where;
∑đ?‘˜đ?‘–=1 đ?‘›đ?‘– = đ?‘›
2) Combination: selection of r objects from n without regard to order
đ?‘›đ??śđ?‘&#x; =
đ?‘›! đ?‘&#x;! (đ?‘› − đ?‘&#x;)!
Properties of a Probability Function 1. đ?‘ƒ(đ?œ™) = 0 2. đ?‘ƒ(đ??´đ?‘? ) = 1 − đ?‘ƒ(đ??´) 3. đ?‘ƒ(đ??´ ∊ đ??ľâ€˛ ) = đ?‘ƒ(đ?‘Ž − đ??ľ) = đ?‘ƒ(đ??´) = đ?‘ƒ(đ??´đ??ľ) 4. đ?‘ƒ(đ??´ âˆŞ đ??ľ) = đ?‘ƒ(đ??´) + đ?‘ƒ(đ??ľ) − đ?‘ƒ(đ??´ ∊ đ??ľ) Conditional Probability
đ?‘ƒ(đ??´|đ??ľ) =
đ?‘ƒ(đ??´đ??ľ) đ?‘ƒ(đ??ľ)
,
đ?‘ƒ(đ??ľ) > 0
nanyari na hinahanapan mo ng probability ďƒ˜ Properties: đ?‘ƒ(đ??´đ??ś|đ??ľ) = 1 − đ?‘ƒ(đ??´|đ??ľ) đ?‘ƒ(đ?œ™|đ??ľ) = 0 đ?‘ƒ(đ??´1 âˆŞ đ??´2 âˆŞ . . .âˆŞ đ??´đ?‘› |đ??ľ = đ?‘ƒ(đ??´1 |đ??ľ) + đ?‘ƒ(đ??´2 |đ??ľ)+. . . +đ?‘ƒ(đ??´đ?‘› |đ??ľ) - for mutually exclusive events đ?‘ƒ(đ??´1 |đ??ľ) = đ?‘ƒ(đ??´1 đ??´2|đ??ľ) + đ?‘ƒ(đ??´1 đ??´2 đ??ś|đ??ľ) đ?‘ƒ(đ??´ âˆŞ đ??´2 |đ??ľ) = đ?‘ƒ(đ??´1 |đ??ľ) + đ?‘ƒ(đ??´2 |đ??ľ) − đ?‘ƒ(đ??´1 đ??´2 |đ??ľ) Properties of Total Probability (for mutually exclusive events) -> B are ME ďƒ˜ đ?‘ƒ(đ??´) = đ?‘ƒ(đ??´|đ??ľ1 )đ?‘ƒ(đ??ľ1 ) + đ?‘ƒ(đ??´|đ??ľ2 )đ?‘ƒ(đ??ľ2 )+. . . +đ?‘ƒ(đ??´|đ??ľđ?‘› )đ?‘ƒ(đ??ľđ?‘› ) ďƒ˜ _______ : đ?‘ƒ(đ??´) = đ?‘ƒ(đ??´|đ??ľ)đ?‘ƒ(đ??ľ) + đ?‘ƒ(đ??´|đ??´đ?‘? )đ?‘ƒ(đ??ľđ?‘? ) ______’ Theorem (for events B that are mutually exclusive) đ?‘ƒ(đ??´|đ??ľđ?‘˜ )đ?‘ƒ(đ??ľđ?‘˜ )
đ?‘ƒ(đ??ľđ?‘˜ |đ??´) = ∑đ?‘›
đ?‘—=1 đ?‘ƒ(đ??´|đ??ľđ?‘— )đ?‘ƒ(đ??ľđ?‘— )
TTP
Multiplication Rule đ?‘› ďƒ˜ đ?‘ƒ(đ?‘›đ?‘–=1 đ??´1 ) = đ?‘ƒ(đ??´1 )đ?‘ƒ(đ??´2 |đ??´1 )đ?‘ƒ(đ??´3 |đ??´1 đ??´2 ). . . đ?‘ƒ(đ??´đ?‘› |đ??´1 đ??´2 . . . đ??´đ?‘› − 1) Ex: đ?‘ƒ(đ??´1 đ??´2 đ??´3 đ??´4 đ??´5 ) = đ?‘ƒ(đ??´5 |đ??´1 đ??´2 đ??´3 đ??´4 )đ?‘ƒ(đ??´4 |đ??´1 đ??´2 đ??´3 )đ?‘ƒ(đ??´3 |đ??´1 đ??´2 )đ?‘ƒ(đ??´2 |đ??´1 )đ?‘ƒ(đ??´1 ) ďƒ˜ ________: đ?‘ƒ(đ??´ ∊ đ??ľ) = đ?‘ƒ(đ??´)đ?‘ƒ(đ??ľ|đ??´) = đ?‘ƒ(đ??ľ)đ?‘ƒ(đ??´|đ??ľ) 2
Independent Events ďƒ˜ Must satisfy at least one of the ff: a) đ?‘ƒ(đ??´|đ??ľ) = đ?‘ƒ(đ??´) if đ?‘ƒ(đ??ľ) > 0 b) đ?‘ƒ(đ??ľ|đ??´) = đ?‘ƒ(đ??ľ) if đ?‘ƒ(đ??´) > 0 c) đ?‘ƒ(đ??´đ??ľ) = đ?‘ƒ(đ??´)đ?‘ƒ(đ??ľ) ďƒ˜ Remarks 1) For an event to be mutually exclusive (ME) and independent, at least one of the events should have zero probability 2) If A and B are ME event and both have nonzero probabilities then they cannot be independent 3) If A and B are independent event and both have nonzero probabilities then they cannot be ME 4) If there are more than 2 events, pair wire independence does not imply independence of all events
CREDITS: Notes by Camille Salazar Encoded by Gerald Roy CampaĂąano
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