Standard normal distribution

A normal distribution with:

mean μ=0
variance σ²=1.


General normal random variable can be standardized to a Standard Normal random variable as:

Z = (X - μ) / σ

Confidence Interval for normal distribution

• 68% of the area within ± 1σ (standard deviations of mean), i.e. 68% confidence interval =X_a ± 1σ
• 90% of the area within ± 1.645 σ
• 95% of the area within ± 1.96 σ
• 99% of the area within ± 2.58

Properties of normal distribution

Properties:

• A continuous, symmetric probability distribution (bell-shaped)
• Mean located at its central point.
• Linear combination of normal variables is also normal.
• Completely defined by mean and standard deviation.
• Skewness = 0 , i.e. mean = median = mode.
• Kurtosis = 3 and excess kurtosis = 0
• Range from -infinity to +infinity, the probability outside ± 3 σ is extremely low

Unconditional probabilities vs. Conditional probability

Unconditional probabilities
An independent chance that a single outcome results from a sample of possible outcomes, without reference to any other event.

P(A) = no of time of “A” occur / total no of possible outcomes

Conditional probability (or Mariginal probability)
Probability of an event or outcome is based on the occurrence of a previous event or outcome. Conditional probabilities are important in tests of market efficiency, where event B is some piece of public or private information that becomes available to the market at some point of time.

Probability of A if B occurred,

P(A│B) = Joint probability of A and B / Unconditional probability of B
= P(AB) / P(B),

where: P(B) is not equal to 0

Inconsistent probabilities

Two assets are priced on the basis of probabilities that are different but are assigned to the same event, i.e. overpriced or underpriced. This inconsistence triggers buy and sell that should then elimiate the inconsistence.

Probability of an event in terms of odds for or against the event

Given a probability P(E),

Odds "FOR",

E = P(E)/[1 – P(E)]

e.g. A probability of 20% would be "1 to 4".


Odds "AGAINST",

E = [1 – P(E)]/P(E)

e.g.A probability of 20% would be "4 to 1".

Objective probability vs. Subjective probability

Objective probability
The estimates should not vary from person to person. Emprirical and a priori probabilities are sometimes referred to as objective probabilities.

Subjective probability
Derive from an individual's personal judgment and contain no formal calculations and only reflect the subject's opinions and past experience. It is the least formal way.

Empirical probability vs. A priori probability

Empirical probability
The probability of an event occurring is estimated from data, usually in the form of a relative frequency.

A priori probability
The probability of an event is deduced by reasoning about the structure of the problem itself.

Properties of probability

For any event i,

0≦P(E_i)≦1

If E₁..E_n are mutually exclusive and exhaustive, then

ΣP(E_i)=1

Mutually exclusive and Exhaustive

Mutually exclusive
Only one event occur at a time.

Exhaustive
The set of events includes all possible outcomes.

Radom Variable, Outcome, Event

The basic concepts of statistics and probability theory are essential to describe the main statistical properties of a population and apply in various probability concepts in practice. Probability is used to measure risk.

Random Variable
Uncertain quantity arising from a random experiment.

Outcome
Outcome refers to any possible value that a random variable can take.

An event
A specified outcome from a random experiment