# CFA – Ethical & Professional Standards & Quantitative Methods – Probability Concepts

# 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

- The sum of probabilities of all possible mutually exclusive events is 1
- 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

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^{i }

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%.