Reading 3: Probability Concepts Flashcards
Define a (1) random variable, (2) an outcome, and (3) an event.
- A random variable is an uncertain value determined by chance.
- An outcome is the realization of a random variable.
- An event is a set of one or ore outcomes. Two events that cannot both occur are termed “mutually exclusive”, and a set of events that includes all possible outcomes is an “exhaustive” set of events.
Identify the two defining properties of probability, includign mutually exclusive and exhaustive events, and compare and contrast empirical, subjective, and a priori probabilities.
The two priorities of probability are as follows:
- The sum of the probabilities of all possible mutually exclusive events is 1.
- The probability of any event cannot be greater than 1 or less than 0.
A priori probability measures. predetermined probabilities based on well-defined inputs, empirical probability measures probability from observations or experiments, and subjective probability is an informed guess.
Describe the probability of an event in terms of odds for and against the event.
Probabilities can be stated as odds that an event will or will not occur. If the probability of an event is A out of B trials (A/B) the “odd for” are A to (B-A) and the “odds against” are (B-A) to A.
Calculate and interpret conditional probabilities.
- Unconditional probability (marginal probability) is the probability of an event occurring.
- Conditional probability, P(A|B), is the probability of an event A occuring given that event B has occured.
Demonstrate the application of the multiplication and addition rules for probability.
The multiplication rule of probability is used to determine the joint probability of two events.
P(AB) = P(A | B) * P(B)
The addition rule of probability is used to determine the probability that at least one of two events will occur:
P(A or B) = P(A) + P(B) - P(AB)
The total probability rule is used to determine the unconditional probability of an event, given conditional probabilities:
P(A) = P(A | B1) * P(B1) + P(A | B2) * P(B2) + … + P(A | BN) * P(BN)
Where B1, B2, … , BN is a mutually exclusive and exhaustive set of outcomes.
Compare and contrast dependent and independent events.
The probability of an independent event is unaffected by the occurrence of other events, but the probability of a dependent event is changed by the occcurrence of another event. Events A and B are independent only if:
P(A | B) = P(A), or equivalently, P( B | A) = P(B)
Calculate and interpret and unconditional probability using the total probability rule.
Using the total probability rule, the unconditonal proobability of A is the probability-weighted sum of the conditional probabilities:
P(A) = Σ[Pi(Bi)] * P(A | Bi)
where Bi is a set of mutually exclusive and exhaustive events.
Calculate and interpret the expected value, variance, and standard deviation of random variables.
The expected value of a random variable is the weighted average of its possible outcomes:
E(X) = ΣP(xi)xi = P(x1)x1 + P(x2)x2 + … + P(xn)xn
Variance can be calculate as the probability-weighted sum of the squared deviations from the mean or expected value. The standard deviation is the positive square root of the variance.
Explain the use of the conditional expectation in investment applications.
Conditional expected values depend on the outcomes of some other event.
Forecast of expexted values for a stock’s returns, earnings, and dividends can be refined, using conditionals expected values, when new information arrives that affects the expected outcomes.
Interpret the probability tree and demonstrate its application to investment problems.
A probability tree shows the probabilities of two events and the conditional probabilities of two subsequent events.
Calculate and interpret the expexted value, variance, standard deviation, covariances and correlations of portfolio returns.
Calculate and intepret the covariance of portfolios returns using the joint probability function.
Given the joint probabilities for Xi and Yi, i.e., P(Xi Yi) the covariance is calculated as:
ΣP(Xi Yi) [Xi- E(X)] [Yi - E(Y)]
Calculate and interpret an updated probability using Baye’s formula.
Identify the most appropriate method to solve a particular counting problem and analyze counting problems using factorial, combination, and permutation concepts.
