CFA Level 1 - Reading 3. Probability Concepts

STUDY SESSION 1 - QUANTITATIVE METHODS

READING 3 - Probability Concepts

 

3.1. Conditional and Joint Probabilities

Terms	(Random variable) - Uncertain quantity or number. (Outcome) - Observed value of a random variable. (Event) - Single outcome or a set of outcomes. (Mutually exclusive events) – Events that cannot happen at the same time. (Exhaustive events) – Events that include all possible outcomes.
Properties of Probability	- Probability of occurrence of any event (Ei) is between 0 and 1 - If a set of events is mutually exclusive and exhaustive, the probabilities of those events sum to 1.
Objective Probability	(Empirical Probability) –Established by analyzing past data or outcome. (Priori Probability) – Determined using a formal reasoning and inspection process(not data)
Subjective Probability	The least formal method of developing probabilities and involve the use of personal judgement.
Odds	The odds of an event represent the ratio of the (probability that the event will occur) / (probability that the event will not occur).
Unconditional Probability	The probability of an event regardless of the past or future occurrence of other events.
Conditional Probability	The occurrence of one event affects the probability of the occurrence of another event. * The probability of A given the occurrence of B. * A conditional probability of an occurrence is also called its likelihood.
Joint Probability	The probability that the two events will both occur. * Also referred as the multiplication rule of probability.
Addition Rule of Probability	The probability that at least one of two events will occur. * B1, B2, … Bn is a mutually exclusive and exhaustive set of outcomes.
Joint Probability of Independent Events	The multiplication rule is used to calculate the joint probability of any number of independent events.

 

3.2. Conditional Expectations and Expected Value

Independent Events	The occurrence of independent events has no influence on the occurrence of the others.
Total Probability Rule	Used to determine the unconditional probability of an event, given conditional probability. * B1, B2, … Bn is a mutually exclusive and exhaustive set of outcomes. or * Bc is read “the complement of B”, meaning “not B” * P(Bc) = 1 - P(B)
Expected Value of Random Variable	The weighted average of the possible outcomes for the variable.
Variance and Standard Deviation	Variance and standard deviation measure the dispersion of a random variable around its expected value.   (Variance) : Calculated as the probability-weighted sum of the squared deviations from the mean(or expected value). (Standard Deviation) : Positive square root of the variance

 

3.3. Portfolio Variance, Bayes, and Counting Problems

Portfolio Expected Return	The expected return of a portfolio composed of n assets with weights(w) and expected returns(R).
Covariance	Measure of how two assets move together. * It is the expected value of the product of the deviations of the two random variables from their respective expected values. * Covariance of a random variable with itself is its variance of RA : Cov(RA,RA) = Var(RA)
Portfolio Variance	The asset weights, returns variances, and returns covariances are used to calculate the variance of portfolio returns.
Correlation	Correlation matrix to calculate portfolio returns variance.
Bayes' Formula	Used to update a given set of prior probabilities for a given event in response to the arrival of new information.
Labeling Formula	Refers to the situation where there are n items that can each receive one of k different labels.
Combination Formula	The general formula for the two groups of label. * Number of possible ways of selecting r items from a set of n items * the order of selection is not important
Permutation Formula	Different groups of size r in specific order from n objects.
Counting method	* The multiplication rule of counting is used when there are two or mor groups. * Factorial is used by itself when there are no groups. * The labeling formula applies to three or more subgroups of predetermined size. * The combination formula applies to only two groups of predetermined size where the order of selection is not important. * The permutation formula applies to only two groups of predetermined size where the order of selection is important.