Reading 3: Probabilty Concepts Flashcards
Define a random variable, an outcome, and an event
Random Variable: is a quantity whose future outcomes are uncertain
Outcome: is a possible value of a random variable
Event: is a specified set of outcomes
Identify the two defining properties of probability, including mutually exclusive and exhaustive events. Compare and contrast empirical, subjective, and priori probabilities.
The two defining probabilities of probability are: 1) probability of any event is a number between 0 and 1, and 2) sum of the probabilities of any set of mutually exclusive and exhaustive events equals 1.
Mutually exclusive: only once event can occur at a time
Exhaustive Events: that the events cover all possible outcomes
Empirical probability: estimate of the probability of an event as a relative frequency of occurrence based on historical data
Subjective probability: personal assessment of probability without reference to any particular data. Drawing on personal or subjective judgement.
Priori probability: the resulting probability based on logical analysis rather than on observation or personal judgement.
Describe the probability of an event in terms of odds for and against the event
A probability is the function of the time you expect an event to occur and the odds for an event is the probability that an event will occur divided by the probability that the event will not occur.
Dutch Book Theorem: inconsistent probabilities create profit opportunities
Calculate and interpret conditional probabilities
Unconditional Probability: P(A), answers the question of “What is the probability of this event A?”
Conditional Probability: the conditional probability of A given that B has occurred is equal to the joint probability of A and B divide by the probability of B (assuming not equal to 0)
Joint Probability: P(AB), is the probability of both A and B happening
Demonstrate the application of the multiplication and addition rules for probability
Multiplication Rule: when we know the conditional probability and want to know the joint probability, we utilize the multiplication rule; P(AB) = P(A|B)P(B)
Addition Rule: Given events A and B, the probability of A or B occurs or both occur is equal to the probability that A occurs, plus the probability that B occurs, minus the probability that both A and B occur; P(A or B) = P(A) + P(B) - P(AB). Note, if the two events are mutually exclusive, the formula reduces to P(A or B) = P(A) + P(B).
Compare and contrast dependent and independent events
Two events are independent if the occurrence one event does not affect that probability of occurrence of the other event. In short, independence means that knowing B does not infer anything about A, or vice versa.
When two events are not independent, they are dependent, which is the probability of occurrence of one is related to the occurrence of the other.
Multiplication rule for independent events: when two events are both independents, the joint probability is equal to the product of the individual probabilities of A and B ~ P(AB) = P(A)P(B)
Calculate and interpret an unconditional probability using the total probability rule
When the scenarios (conditioning events) are mutually exclusive and exhaustive. We can then analyze the event utilizing the Total Probability Rule, which is a rule which explains the unconditional probability of the event in terms of probabilities conditional on the scenarios. P(A) = P(A|S1)P(S1) + P(A|S2)P(S2) …+P(A|Sn)P(Sn)
Calculate and interpret the expected value, variance, and standard deviation of random variables
Expected Value: probability weighted mean of the possible outcomes of the random variable, denoted E(x).
Variance: Expected Alex of the squared deviations from the random variables expected value
Standard Deviation: Positive square root of the variance
Explain the use of conditional expectation in investment application
When we refine our expectation or forecast , we are typically making adjustment based on new information or events. In these cases, we are utilizing conditional expected values.
Parallel to the Total Probability Rule for stating unconditional probabilities in terms of conditional probabilities, there is a principle for stating unconditional expected values in terms of conditional expected values. This principle is known as the Total Probability Rule for Expected Values: E(X) = E(X|S1)P(S1) + E(X|S2)P(S2) …. + E(X|Sn)P(Sn), where S1, S2, and Sn are mutually exclusive and exhaustive scenarios and events.
Calculate and interpret the expected value, variance, standard deviation, covariances, and correlations of portfolio returns.
Portfolio Expected Return: E(RP) = W1E(R1) + W2E(R2)+ ….+WnE(Rn)
Covariance Sample: Cov(Ri, Rj) = Sum(Ri,t - Ri)(Rj,t - Rj) / (n-1); Captures how the co-movements of returns affect portfolio variance. A negative number, alludes to a inverse relationship, zero, means unrelated, and positive, means its a positive relationship.
Variance: E [ (Rp - E(Rp))^2]
Standard Deviation: Positive square root of the variance
Note: a complete list of the variances constitutes all the statistical data needed to compute portfolio variance of return as a covariance matrix. In general, for n securities, there are n(n-1)/2 distinct covariances and N variances to estimate.
Correlation: between two random variables Ri and Rj, is defined as p(Ri, Rj) = Cov (Ri, Rj) / STD(Ri) x STD(Rj). Note, an alternative notion is Corr (Ri, Rj) or pi,j.
Calculate and interpret the covariance of portfolio returns using the joint probability function
You can utilize the multiplication rule for expected value of the product of uncorrelated random variables, thus the expected value of the product of uncorrelated random variables is the product of their expected values.
Calculate and Interpret an updated probability using Bayes Formula
Bayes Formula: rational method for adjusting our viewpoints as we confront new information. Bayes Formula makes use of the Total Probability rule. The formula works in reverse; more precisely, it reverses the “given that” information. Bayes Formula uses the occurrence of the event to infer the probability of the scenario generating it. For that reason, Bayes Formula is sometimes called an inverse probability.
P(Event|Information) = P(Information|Event) / P(Information) x P(Event)
Conditional probabilities of an observation are sometimes referred to as likelihoods
An updated probability is a called your posterior probability, because it reflects or comes after the new information
When the prior probability are equal, the probability of information given an event equals the probability of the events given the information. When the decision maker has equal prior probabilities (called diffuse priors), the probability of an event is determined by the information.
Identify the most appropriate method to solve a particular counting problem and analyze counting probes using factorial, combination, and permutation concepts.
A number of ways we can assign and divide a task would be ~ N Factorial - n! = n (n-1) (n-2) (n-3)…
Multinomial Formula (General formula for labeling problems): the number of ways that n objects can be labeled with K different labels, with n, of the first type, n2 of the second type, and so with n1 plus n2 plus ….no. This is given by: n!/n!n2!…nk!
Combination Formula: When the order of the list does not matter. The number of ways that we can choose r objects from a total of n objects, when the order in which the r objects are listed does not matter is - nCr.
Permutation: the number of ways that we can choose r objects from a total of n objects, when the order in which the r objects are listed does matter is - nPr.
