Reading 9: Probability Concepts Flashcards
Random Variable
Uncertain Quality or Number
Outcome
The Observed Value of a Random Variable
Event
Single Outcome or Set of Outcomes
Mutually Exclusive Events
Events that cannot both happen at the same time
Exhaustive Events
Events that include all possible outcomes
Two defining properties of Probability
- Probability of the occurrence of any event is between 0 -1
- If a set of events are mutually exclusive and exhaustive, the probability of those events will sum to 1.
Empirical Probability
Established by analyzing past data (observations/experiments)
(Objective Probability)
Priori Probability
Determined using formal reasoning and an inspection process (well-defined inputs)
(Objective Probability)
Subjective Probability
Involves the use of personal judgement. Least formal method of developing probabilities
(Informed guess)
Odds For
A to (B - A)
Odds Against
(B-A) to A
Unconditional Probability
Aka: Marginal Probability
Probability of an event occurring
Uses the Total Probability Rule
Conditional Probabilty
Aka: “Given” / Likelihood
Probability of an event A occurring given that event B has occured
P(A | B)
Where:
“|” = Given
Multiplication Rule of Probability
Used to determine the joint probability of two events
P(AB) = P(A | B) x P(B)
Additional Rule of Probability
Used to determine the probability that at least one of two events will occur
P(A or B) = P(A) + P(B) - P(AB)
NB: if mutually exclusive, then it is simply A + B
Total Probability Rule
Used to determine the unconditional probability of an event, given conditional probabilities
P(A) = P(A | B1)P(B1) + P(A | B2)P(B2)
NB: B1 etc. are mutually exclusive, exhaustive set of outcomes
Joint Probability
Multiplication Rule of Probability
Probability that both events will occur
Independent Events (Definition)
Events where the occurrence of one events has no influence on the occurrence of the others
P(A | B) = P(A) or P(B | A) = P(B)
So P(AB) = P(A) x P(B)
If this condition is not satisfied, the event are dependent e.g. if P(A) > P(A | B)
Expected Value
Weighted average of the possible outcomes of a random variable, where weights are the probabilities that the outcomes will occur
Expected Value (Formula)
P(x1)x1 + p(x2)(x2) etc.
Where:
P(x) = Probability of x
x1 = value
Conditional Expectation
Conditional Expected Values are contingent upon the outcome of some other event
Analyst would use a conditional expected value to revise his expectations when new information arrives
Tree Diagram
Used to show the probabilities of two events and the conditional probabilities of two subsequent events
NB: Probabilities of all possible outcomes should sum to zero
Covariance [Cov(X,Y)]
Measure of how two assets move together
Expected value of the product of their deviations from their respective expected values
Covariance (Formula)
Cov (A,B) = Pi(Ai - EVa)(Bi - EVb)