Probability Concepts Flashcards
Random Variable (RV)
An uncertain value determined by chance
Outcome
The realization of a random variable
Event
A set of one or more outcomes
Mutually Exclusive Event
two events that cannot both occur
Exhaustive Event
Set of events that includes all possible outcomes
Probability Properties
- 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
Priori Probability
Measures predetermined probabilities based on well-defined inputs
Empirical Probability
Measures probability from observations or experiments
Subjective Probability
An informed guess
Unconditional Probability
Marginal Probability. Probability of an event occurring
Conditional Probability
P(A | B). Probability of an event A occurring given that event B has occurred
Joint Probability
P(AB). The probability that two events will both occur. For independent events, P(A|B) = P(A) so that P(AB) = P(A) x P(B). P(AB) for any number of independent events is the product of their individual probabilities. Mutually exclusive events make P(AB) = 0
Independent Events
Events A and B are independent iff:
P(A|B) = P(A) or equivalently, P(B|A) = P(B)
Covariance
Measures the extent to which two random variables tend to be above and below their respected means for each joint realization.
Correlation
Standardized measure of association between two random variables. Ranges from -1 to +1
Spurious Correlation
May result by chance from the relationships of two variables to a third variable
Probability distribution
Lists all the possible outcomes of an experiment, along with their associated probabilities
Discrete Random Variable
Has positive probabilities associated with a finite number of outcomes
Continuous Random Variable
Has positive probabilities associated with a range of outcome values- the probability of any single value is zero
Cumulative Distribution Function (CDF)
Gives the probability that a random variable will be less than or equal to specific values. Expressed as:
F(x) = P(X <= x). Represented by the area under the probability distribution to the left of that value
Discrete Uniform Distribution
n discrete, equally likely outcomes. Probability of each outcome is 1/n
Binomial Distribution
Probability distribution for a binomial (discrete) random variable that has two possible outcomes
Normal Probability Distribution Characteristics
- Normal curve is symmetrical and bell-shaped with a single peak at the exact center of the distribution
- Mean = median = mode and all are in the exact center of the distribution
- Normal Distribution can be completely defined by its mean standard deviation b/c the skew is always 0 and kurtosis is always 3
Multivariate Distributions
Describe the probabilities for more than one random variable, whereas a univariate distribution is for a single random variable. Correlation for multivariate distribution describes the relation between the outcomes of its variables relative to their expected values
Confidence Interval
A range within which we have a given level of confidence of finding a point estimate
Confidence intervals for a normally distributed RV
90%: u +/- 1.65 std. dev.
95%: u +/- 1.96 std. dev.
99%: u +/- 2.58 std. dev.
Shortfall Risk
Probability that a portfolio’s value (or return) will fall below a specific value over a given period of time. Greater SFR’s are preferred and indicate a smaller shortfall probability. Optimal portfolio minimizes shortfall risk
Lognormal Distribution
often used to model asset prices, since a lognormal RV cannot be negative and can take on any positive value
Monte Carlo Simulation
Uses randomly generated values for risk factors, based on their assumed distributions, to produce a distribution of possible security values. Limitations are that it is fairly complex and will provide answers that are no better than the assumptions used
Historical Simulation
Uses randomly selected past changes in risk factors to generate a distribution of possible security values. Limitation includes that it cannot consider the effects of significant events that did not occur in the sample period
