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)
Covariance (Properties)
- Extent to which two random variables tend to be above and below their respective means for each joint realization
- Covariance of itself is the same as the variance
- Covariance may range from negative infinity to positive infinity
Correlation Coefficient
Standardized measure of association between two random variance; it ranges in value from -1 to +1
Correlation Coefficient (Formula)
Cov (A,B) / SDa x SDb
NB: Remember to convert variances to SDs
Correlation Coefficient (Properties)
Measures the strength of the linear relationship between two random variables
Has no units
Ranges from -1 to +1
Correlation Coefficient = 1
Perfect Positive correlation. When one moves, the other moves in a proportional positive direction relative to its mean
Correlation Coefficient = -1
Perfect Negative correlation. Movement in one variable will result in the exact opposite proportional movement in the other relative to its mean
Correlation Coefficient = 0
No linear relationship between variables
Portfolio Variance Calculation Steps
- Weights (Individual)
- Expected Value (Individual)
- Variances (individuals)
- Coveriance
- Portfolio Variance
Variance of 2-asset portfolio (Formula)
Learn this. Refer to textbook
Bayes’ Formula (Definition)
Used to update a given set of prior probabilities for a given event in response to the arrival of new information
Bayes’ Formula (Formula)
Updated Probability = (Prob of new info / unconditional prob of new info) x prior probability of event
Priors
What is already known
Labeling
Refers to a situation where there are n items that can each receive one of k different lables
Labeling (Formula)
n! / (n1!) x (n2!) x (n3!) etc.
Where:
! = Factoring (e.g. 4! = e x 3 x 3 x 1 = 24)
n! = total
Factoring
! e.g. 3! = 3 x 2 x 1
Combination (or binomial) Formula
General Formula for labeling when k = 2
= n! / (n-r)r!
Number of possible ways (combinations) of selecting r items from a set of n items when the order of selection is not important
Permutation (Definition)
Specific ordering of a group of objects
Permutation (Formula)
n! = (n-r)!
How many different groups of size r in specific order can be chosen from n objects
5 Guidelines for Determining Counting Methods:
- Multiplication Rule of Counting
- Two or more groups and only one item may be selected - Factorial
- When there are no groups - Labeling Formula
- Three or more subgroups of predetermined size - Combination Formula
- Only two groups of predetermined size (‘chose’ or ‘combination’) - Permutation Formula
- Only two groups of predetermined size (‘order’ is important)
