# QM 3 ( Reading 8, Probabilities )

version from 2017-03-22 17:00

## Section

OutcomeThe observed value of a random variable
EventCould be a single outcome, or a set of outcomes
Exhaustive eventsEvents that cover all possible outcomes for an event
Methods of Estimating probabilitiesEmpirical, Subjective, Priori
Empirical probabilityEstimates the probability of an event based on the frequency of its occurance in the past
Subjective probabilityDraws on subjective reasoning and personal judgment to estimate probabilities
Priori Probabilitybased on formal analysis and reasoning rather than personal judgement.
Dutch Book Theoremif Probabilities reflected in the stock prices are not consistent, they give rise to profit opportunities until it is resolved
Unconditional ProbabilityProbability whose occurence does not rely on another happening and is a stand alone probability
Conditional Probability ( P(A|B) )Probability of an event occurring given that another event has happened.
Joint probabilityProbability of both A and B happening (P (AB) ). if A and B are mutually exclusive, both of them cannot happen at the same time and hence, P(AB)=0
P(A|B) formulaP(A|B)= P(AB)/P(B) ===> P(AB) = P(A|B) x P(B)
P(A or B) formulaP ( A or B ) = P(A) + P(B) - P(AB)
Dependent eventEvents whose occurrence depends on other events
independent eventevents whose occurrence does not depend on another event.
P(A|B) for independent eventsP(A|B)= P(A) & P(B|A) = P(B)
P(A and B) for independent events = P(A) x P(B)
Total probability ruleExpress the unconditional probability of an event in terms of conditional probabilities for mutually exclusive and exhaustive events.
Total probability FormulaP(A) = P(A|S1) x P(S1) + P(A|S2) x P(S2) +...+ P(A|Sn) x P(Sn)
Expected ValueProbability weighted average of all possible outcomes
Expected value FormulaE(X) = Sigma P(X)Xi ==> for a dice, its 1/6 x 1 + 1/6x2 + 1/6x3 + ...
Variance of XE{[X-E(X)]^2}
Variance of XSigma P(Xi)[Xi - E(X)]^2\
Unconditional Value based on Conditional valuesE(X) = E (X|S)P(S) + E (X|S^)P(S^)
Unconditional Value based on Conditional valuesE(X) = E (X|S1)P(S1) + E (X|S2)P(S2) + .... + E (X|Sn) P(Sn)