Find The Right Fit With Probability Distributions
By David Harper
Probability Distributions are pictures that describe a particular view of uncertainty.
An emergent research view holds that financial markets are both uncertain and predictable. Also, markets can be efficient but also uncertain. In finance, we use probability distributions to draw pictures that illustrate our view of an asset return's sensitivity when we think the asset return can be considered a random variable.
There are two ways of categorizing distributions: by whether it is discrete or continuous, and by whether it is a probability density function (PDF) or a cumulative distribution.
The PDF is P[x=X]; The cumulative distribution is P[x<=X]
UniformThe simplest and most popular distribution is the uniform distribution in which all outcomes have an equal chance of occurring.
UniformThe simplest and most popular distribution is the uniform distribution in which all outcomes have an equal chance of occurring.
Now roll three dice together, as shown in Figure 4. We start to see the effects of a most amazing theorem: the central limit theorem.
The central limit theorem boldly promises that the sum or average of a series of independent variables will tend to become normally distributed, regardless of their own distribution. Our dice are individually uniform but combine them and - as we add more dice - almost magically their sum will tend toward the familiar normal distribution!
The central limit theorem boldly promises that the sum or average of a series of independent variables will tend to become normally distributed, regardless of their own distribution. Our dice are individually uniform but combine them and - as we add more dice - almost magically their sum will tend toward the familiar normal distribution!
Binomial
The binomial distribution reflects a series of "either/or" trials, such as a series of coin tosses. These are called Bernoulli trials but you don't need even (50/50) odds. A Bernoulli trial refers to events that have only two outcomes.
The binomial distribution below plots a series of 10 coin tosses where the probability of heads is 50% (p-0.5). You can see in Figure 6 that the chance of flipping exactly five heads and five tails (order doesn't matter) is just shy of 25%:
The binomial distribution reflects a series of "either/or" trials, such as a series of coin tosses. These are called Bernoulli trials but you don't need even (50/50) odds. A Bernoulli trial refers to events that have only two outcomes.
The binomial distribution below plots a series of 10 coin tosses where the probability of heads is 50% (p-0.5). You can see in Figure 6 that the chance of flipping exactly five heads and five tails (order doesn't matter) is just shy of 25%:
As the number of trials increase, the binomial tends toward the normal distribution.
LognormalThe lognormal distribution is very important in finance because many of the most popular models assume that stock prices are distributed lognormally. It is easy to confuse asset returns with price levels:
http://www.investopedia.com/articles/06/probabilitydistribution.asp
LognormalThe lognormal distribution is very important in finance because many of the most popular models assume that stock prices are distributed lognormally. It is easy to confuse asset returns with price levels:
http://www.investopedia.com/articles/06/probabilitydistribution.asp
