Probability concepts Flashcards
Random variable
quantity with uncertain outcomes
Outcome
possible values
event
specified set of outcomes
Mutually exclusive events
Only one event can occur at a time
Exhaustive events
Events cover all possible outcomes
Probability
number between 0 and 1 that measures chance event will occur.
empirical vs. subjective vs. a priori probabilities
Empirical - probability estimate using relative frequency based on historical data
Subjective - personal judgment of probability
A priori - probability based on logical analysis
state probability as odds for and against
for = a/a+b against = b/a+b
unconditional vs. conditional probabilities
Unconditional - what is probability of event X
Conditional - what is probability of A given that B occurred
P(A | B) is “probability of A given B”
P(A | B) = P(AB)/P(B)
multiplication, addition, and total probability rules
Multiplication: joint probability of two events
Addition: probability at least one of two events occur
Total: unconditional probability of event in terms of probabilities conditional on scenarios
calculate and interpret joint probability of two events
P(AB) = P(A | B) * P(B)
calculate and interpret probability that at least one of two events will occur
P(A or B) = P(A) + P(B) - P(AB)
calculate and interpret joint probability of independent events
P(AB) = P(A)P(B)
dependent vs. independent evnets
Occurrence of A does not depend on B and vice-a-versa.
calculate and interpret unconditional probability using total probability rule
P(A) = P(A | S1)P(S1) + P(A | S2)P(S2) + … P (A|Sn)P(Sn)
explain use of conditional expectation in investment
Add up possible expected values given mutually exclusive and exhaustive scenarios. Update with new info.
explain tree diagram
Nodes are different branches. Each decision point has 1 total probability. Can add up expected value based on weighted probabilities of each outcome.
calculate and interpret covariance and correlation
Cov(Ri,Rj) = E [(Ri - ERi)(Rj - ERj)]
Make Covariance matrix.
Cov(R1,R2,R3) = w1^2 * (varR1) + w2^2 * (varR2) + w3^2 * (varR3) + 2w1w2cov(R1,R2) + 2w1w3cov(R1,R3) + 2w2w3ccov(R2,R3)
Corr(Ri,Rj) = Cov(Ri,Rj) / σ(Ri) * σ(Rj) σ = square root of covariance
0 - uncorrelated
Positive - positive linear relationship
Negative - inverse linear relationship
calculate and interpret expected value, variance, standard deviation
E(X) = Σ P(Xi)(Xi)
σ^2 = Σ P(Xi) [Xi - E(X)]^2
Std deviation: square root of variance (σ)
calculate and interpret covariance given joint probability
Make table of joint probabilities and multiply each joint probability by the respective expected value. Result is the expected return of each asset.
calculate and interpret updated probability using Bayes’ formula
In light of new info, what is updated probability of event?
P (Event | Info) = P(Info | Event) / P(Info) * P(Event)
Write out conditional probabilities for Info given all Events.
P(Info) = Σ P(Info | Event) * P(Event)
identify best method to solve counting problem
If infinite outcomes, no tool.
If assign each member to one slot, use factorial.
If want to count r objects from n and order doesn’t matter, use combination formula
If want to count number of ways choose r objects from n and order matters, use permutation formula
Solve counting problems using factorial, combination, and permutation concepts
Factorial: n! (on calc)
Combination: nCr = n! / (n-r)! r!
Permutation: nPr = n! / (n-r)!
