LOS 8a

A random variable is an uncertain value determined by chance

An outcome is the realization of a random variable

An event is a set of one or more outcomes.  Two events that cannot both occur are termed “mutually exclusive”
and a set of events that includes all possible outcomes is an “exhaustive” set of events.

LOS 8b

The 2 properties of probabilities are

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

A priori probability measures probabilities based on well defined inputs; empirical probability measures probability from observations or experiments; and subjective probability is an informed guess.

LOS 8c

If the probability of an event is A out of B trials (A/B), the ‘odds for’ are A to (B-A) and the ‘odds against’ are (B-A) to A

LOS 8d

Unconditional probability (marginal probability)  is the probability of an event occurring; conditional probability, P(A|B), is the probability of an event (A) occurring given that another event (B) has occurred.

LOS 8e

The joint probability of 2 events, P(A|B), is the probability that they will both occur.

P(A|B), is the probability of an event (A) occurring given that another event (B) has occurred.

The probability that at least one of two events will occur is P(A or B) = P(A) + P(B) – P(AB).
For a mutually exclusive evens, P(A or B) = P(A) + P(B), since P(AB) = 0

The joint probability of any number of independent events is the product of their individual probabilities.

LOS 8f

The probability of an independent event if unaffected by the occurrence of other events, but the probability of a dependent event is changed by the occurrence of another event.

LOS 8g

Using the total probability rule, the unconditional probability of A is the probability weighted  sum of the conditional probabilities:

LOS 8h

Conditional expectations are used in investments to update expectations when a conditioning event has occurred.

LOS 8i

A tree diagram shows the probabilities of 2 events and the conditional probabilities of 2 subsequent events

LOS 8j

Covariance measures the extent to which 2 random variables tend to be above and below their respective means for each joint realization.  It can be calculated as:

Correlation is a standardised measure of association between 2 random variables; it ranges in value from –1 to +1 and is equal to

Correlation coefficient = Cov A,B / (SD A x SD B)
= -7.2 / (2.450 x 3.098)
= -0.9486

LOS 8K

The expected value of a random variable, E(X) equals

The variance of a random variable Var(X) equals

Standard deviation

The expected returns and variance of a 2 asset portfolio are given by

LOS 8l

Given the joint probablities for Xi & Yi ie  P(XiYi) the covariance is calculated as

where Bi is a set of mutually exclusive and exhaustive events.

LOS 8h

Conditional expectations are used in investments to update expectations when a conditioning event has occurred.

LOS 8i

Bayes formula for updating probabilities based on the occurrence of an event O is:

LOS 8n

The number of ways to order n objects is n factorial n! = n x (n-1) x (n-2) x…. x1.

There are   ways to assign k different labels to N items, where n

is the number of items with the label i

The number of ways to choose a subset of size r from a set of size n when order doesnt matter is

when order matters, there are   permutations.

Notes

The probability distribution of annual returns from investing in Company A is given below.

Return %  Probability

20              0.1

30              0.6

40              0.3

Press 2nd Data. Enter in the following:
X01: 20 Y01: 10
X02: 30 Y02: 60
X03: 40 Y03: 30
You are entering the probabilities as whole numbers in the ‘Y’values e.g. a probability of 0.1 = 10% for Y01.
Now press 2nd STAT. Make sure you have 1-V on the screen.
(If it is not keep presssing, 2nd then ENTER until 1-V appears.)
Scroll down until you find the mean as 32% and the population standard deviation of 6%.