Reading 9: Probability Concepts

Question 1

Random Variable

Answer 1

Uncertain Quality or Number

Question 2

Outcome

Answer 2

The Observed Value of a Random Variable

Question 3

Event

Answer 3

Single Outcome or Set of Outcomes

Question 4

Mutually Exclusive Events

Answer 4

Events that cannot both happen at the same time

Question 5

Exhaustive Events

Answer 5

Events that include all possible outcomes

Question 6

Two defining properties of Probability

Answer 6

1. Probability of the occurrence of any event is between 0 -1
2. If a set of events are mutually exclusive and exhaustive, the probability of those events will sum to 1.

Question 7

Empirical Probability

Answer 7

Established by analyzing past data (observations/experiments)

(Objective Probability)

Question 8

Priori Probability

Answer 8

Determined using formal reasoning and an inspection process (well-defined inputs)

(Objective Probability)

Question 9

Subjective Probability

Answer 9

Involves the use of personal judgement. Least formal method of developing probabilities

(Informed guess)

Question 10

Odds For

Answer 10

A to (B - A)

Question 11

Odds Against

Answer 11

(B-A) to A

Question 12

Unconditional Probability

Answer 12

Aka: Marginal Probability

Probability of an event occurring

Uses the Total Probability Rule

Question 13

Conditional Probabilty

Answer 13

Aka: “Given” / Likelihood

Probability of an event A occurring given that event B has occured

P(A | B)
Where:
“|” = Given

Question 14

Multiplication Rule of Probability

Answer 14

Used to determine the joint probability of two events

P(AB) = P(A | B) x P(B)

Question 15

Additional Rule of Probability

Answer 15

Used to determine the probability that at least one of two events will occur

P(A or B) = P(A) + P(B) - P(AB)

NB: if mutually exclusive, then it is simply A + B

Question 16

Total Probability Rule

Answer 16

Used to determine the unconditional probability of an event, given conditional probabilities

P(A) = P(A | B1)P(B1) + P(A | B2)P(B2)

NB: B1 etc. are mutually exclusive, exhaustive set of outcomes

Question 17

Joint Probability

Answer 17

Multiplication Rule of Probability

Probability that both events will occur

Question 18

Independent Events (Definition)

Answer 18

Events where the occurrence of one events has no influence on the occurrence of the others

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

So P(AB) = P(A) x P(B)

If this condition is not satisfied, the event are dependent e.g. if P(A) > P(A | B)

Question 19

Expected Value

Answer 19

Weighted average of the possible outcomes of a random variable, where weights are the probabilities that the outcomes will occur

Question 20

Expected Value (Formula)

Answer 20

P(x1)x1 + p(x2)(x2) etc.
Where:
P(x) = Probability of x
x1 = value

Question 21

Conditional Expectation

Answer 21

Conditional Expected Values are contingent upon the outcome of some other event

Analyst would use a conditional expected value to revise his expectations when new information arrives

Question 22

Tree Diagram

Answer 22

Used to show the probabilities of two events and the conditional probabilities of two subsequent events

NB: Probabilities of all possible outcomes should sum to zero

Question 23

Covariance [Cov(X,Y)]

Answer 23

Measure of how two assets move together

Expected value of the product of their deviations from their respective expected values

Question 24

Covariance (Formula)

Answer 24

Cov (A,B) = Pi(Ai - EVa)(Bi - EVb)

Question 25

Covariance (Properties)

Answer 25

1. Extent to which two random variables tend to be above and below their respective means for each joint realization
2. Covariance of itself is the same as the variance
3. Covariance may range from negative infinity to positive infinity

Question 26

Correlation Coefficient

Answer 26

Standardized measure of association between two random variance; it ranges in value from -1 to +1

Question 27

Correlation Coefficient (Formula)

Answer 27

Cov (A,B) / SDa x SDb

NB: Remember to convert variances to SDs

Question 28

Correlation Coefficient (Properties)

Answer 28

Measures the strength of the linear relationship between two random variables

Has no units

Ranges from -1 to +1

Question 29

Correlation Coefficient = 1

Answer 29

Perfect Positive correlation. When one moves, the other moves in a proportional positive direction relative to its mean

Question 30

Correlation Coefficient = -1

Answer 30

Perfect Negative correlation. Movement in one variable will result in the exact opposite proportional movement in the other relative to its mean

Question 31

Correlation Coefficient = 0

Answer 31

No linear relationship between variables

Question 32

Portfolio Variance Calculation Steps

Answer 32

1. Weights (Individual)
2. Expected Value (Individual)
3. Variances (individuals)
4. Coveriance
5. Portfolio Variance

Question 33

Variance of 2-asset portfolio (Formula)

Answer 33

Learn this. Refer to textbook

Question 34

Bayes’ Formula (Definition)

Answer 34

Used to update a given set of prior probabilities for a given event in response to the arrival of new information

Question 35

Bayes’ Formula (Formula)

Answer 35

Updated Probability = (Prob of new info / unconditional prob of new info) x prior probability of event

Question 36

Priors

Answer 36

What is already known

Question 37

Labeling

Answer 37

Refers to a situation where there are n items that can each receive one of k different lables

Question 38

Labeling (Formula)

Answer 38

n! / (n1!) x (n2!) x (n3!) etc.
Where:
! = Factoring (e.g. 4! = e x 3 x 3 x 1 = 24)
n! = total

Question 39

Factoring

Answer 39

! e.g. 3! = 3 x 2 x 1

Question 40

Combination (or binomial) Formula

Answer 40

General Formula for labeling when k = 2

= n! / (n-r)r!

Number of possible ways (combinations) of selecting r items from a set of n items when the order of selection is not important

Question 41

Permutation (Definition)

Answer 41

Specific ordering of a group of objects

Question 42

Permutation (Formula)

Answer 42

n! = (n-r)!

How many different groups of size r in specific order can be chosen from n objects

Question 43

5 Guidelines for Determining Counting Methods:

Answer 43

1. Multiplication Rule of Counting
   - Two or more groups and only one item may be selected
2. Factorial
   - When there are no groups
3. Labeling Formula
   - Three or more subgroups of predetermined size
4. Combination Formula
   - Only two groups of predetermined size (‘chose’ or ‘combination’)
5. Permutation Formula
   - Only two groups of predetermined size (‘order’ is important)