Reading 8: Probability Concepts Flashcards
Definition of Conditional Probability
Probability Concepts
8.1
The conditional probability of A given that B has occured is EQUAL to the joint probability of A and B divided by the probability of B (assumed to not equal 0).
Multiplication Rule for Probability
Probability Concepts
8.2
The joint probability of A and B equals the probability of A given B times the probability of B.
Because P(AB) = P(BA), the expression P(AB) = P(BA) = P(B | A)P(A) is equalent to the equation pictured.
Addition Rule for Probabilities
Probability Concepts
8.3
Given events A and B, the probability that 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.
If A and B are mutually exclusive, P(AB) can be considered 0.
Multiplication Rule for Independent Events
Probability Concepts
8.4
When two events are independent, the joint probability of A and B equals the product of the individual probabilities of A and B.
The Total Probability Rule
Probability Concepts
8.5 and 8.6
When scenarios (conditioning events) are mutually exclusive and exhaustive, no possible outcomes are left out. We can then analyze the event using the total probability rule. This rule explains the unconditional probability of the event in terms of probabilities conditional on the scenarios.
If we have an event or senario S, the event not-S, called the complement of s, is written SC. Note that P(S) + P(SC) = 1, as either S or not-S must occur.
Equation 5 is for two scenarios.
Equation 6 states the rule for the general cas of n mutually exclusive and exhaustive events or scenarios.
Definition of Expected Value
Probability Concepts
8.7
The expected value of a random variable is the probability-weighted average of the possible outcomes of the random variable.
For a random variable X, the expected value of X is denoted E(X)
Definition of Variance (of a random variable)
Probability Concepts
8.8
The variance of a random variable is the expected value (the probability-weighted average) of squared deviations from the random varaible’s expected value.
Calculating Variance Given Expected Value
Probability Concepts
8.9
Where Xi is one of n possible outcomes of the random variable X.
Conditional Expected Values
Probability Concepts
8.10
The expected value of a random variable X given an event or scenario S is denoted E(X | S). Suppose the random variable X can take on any one of n distinct outcomes X1, X2, …, Xn (these outcomes from a set of mutually exclusive and exhaustive events).
The expected value of X conditional on S is the first outcome, X1, times the probability of the first outcome given S, P(X1 | S), plus the second outcome, X2, time the probability of the second outcome given S, P(X2 | S), and so forth.
Total Probability Rule for Expected Value
Probability Concepts
8.11 and 8.12
The second equation states that the expected value of X equals the expected value of X given Scenario 1, E(X | S1), times the probability of Scenario 1, P(S1), plus the expected value of X given Scenario 2, E(X | S2), times the probabiltity of Scenario 2, P(S2), and so forth.
Expected Value of a Weighted Sum of Random Variables
Probability Concepts
8.13
Covariance (of two random variables)
Probability Concepts
8.14
Portfolio Variance
Given covariance (expanded)
Probability Concepts
8.15
Portfolio Variance
Given covariance (condensed)
Probability Concepts
8.15
Covariance Between Random Variables RA and RB
Probability Concepts
8.18
