Probability Concepts

Question 1

Random Variable (RV)

Answer 1

An uncertain value determined by chance

Question 2

Outcome

Answer 2

The realization of a random variable

Question 3

Event

Answer 3

A set of one or more outcomes

Question 4

Mutually Exclusive Event

Answer 4

two events that cannot both occur

Question 5

Exhaustive Event

Answer 5

Set of events that includes all possible outcomes

Question 6

Probability Properties

Answer 6

1. The sum of the probabilities of all possible mutually exclusive events is 1
2. The probability of any event cannot be greater than 1 or less than 0

Question 7

Priori Probability

Answer 7

Measures predetermined probabilities based on well-defined inputs

Question 8

Empirical Probability

Answer 8

Measures probability from observations or experiments

Question 9

Subjective Probability

Answer 9

An informed guess

Question 10

Unconditional Probability

Answer 10

Marginal Probability. Probability of an event occurring

Question 11

Conditional Probability

Answer 11

P(A | B). Probability of an event A occurring given that event B has occurred

Question 12

Joint Probability

Answer 12

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

Question 13

Independent Events

Answer 13

Events A and B are independent iff:

P(A|B) = P(A) or equivalently, P(B|A) = P(B)

Question 14

Covariance

Answer 14

Measures the extent to which two random variables tend to be above and below their respected means for each joint realization.

Question 15

Correlation

Answer 15

Standardized measure of association between two random variables. Ranges from -1 to +1

Question 16

Spurious Correlation

Answer 16

May result by chance from the relationships of two variables to a third variable

Question 17

Probability distribution

Answer 17

Lists all the possible outcomes of an experiment, along with their associated probabilities

Question 18

Discrete Random Variable

Answer 18

Has positive probabilities associated with a finite number of outcomes

Question 19

Continuous Random Variable

Answer 19

Has positive probabilities associated with a range of outcome values- the probability of any single value is zero

Question 20

Cumulative Distribution Function (CDF)

Answer 20

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

Question 21

Discrete Uniform Distribution

Answer 21

n discrete, equally likely outcomes. Probability of each outcome is 1/n

Question 22

Binomial Distribution

Answer 22

Probability distribution for a binomial (discrete) random variable that has two possible outcomes

Question 23

Normal Probability Distribution Characteristics

Answer 23

• 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

Question 24

Multivariate Distributions

Answer 24

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

Question 25

Confidence Interval

Answer 25

A range within which we have a given level of confidence of finding a point estimate

Question 26

Confidence intervals for a normally distributed RV

Answer 26

90%: u +/- 1.65 std. dev.
95%: u +/- 1.96 std. dev.
99%: u +/- 2.58 std. dev.

Question 27

Shortfall Risk

Answer 27

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

Question 28

Lognormal Distribution

Answer 28

often used to model asset prices, since a lognormal RV cannot be negative and can take on any positive value

Question 29

Monte Carlo Simulation

Answer 29

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

Question 30

Historical Simulation

Answer 30

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