Reading 6/7/2010 (MATHS)

1. Under the efficient market hypothesis (currently the market reflects all available information and the future information can not be predicted especially in very small intervals) and the central limit theorem under continuity assumption, we can show the stock price follows log-normal distribution.  If the continuity assumption is rejected, we can have jump diffusion models.

2.   The probability density function (pdf) of the chi-square distribution is

where Γ(k/2) denotes the Gamma function, which has closed-form values at the half-integers.

It is very interesting as the sum of squares of n independent standard random variables has the chi-square distribution  of n-1 freedom.

3. Student's t distribution 

Student's t-distribution is the probability distribution of the ratio

where

• Z is normally distributed with expected value 0 and variance 1;
• V has a chi-square distribution with ν degrees of freedom;
• Z and V are independent.

• Student's t-distribution has the probability density function

  

  where ν is the number of degrees of freedom and Γ is the Gamma function.

  
  

  Asymptotically , if ν  goes to infinity (sample length increases), the distribution gives to a normal distribution.

  
  

  4. F distribution 

  A random variate of the F-distribution arises as the ratio of two chi-squared variates:

  

  where

  • U₁ and U₂ have chi-square distributions with d₁ and d₂ degrees of freedom respectively, and
  • U₁ and U₂ are independent (see Cochran's theorem for an application).

  The probability density function of an F(d₁, d₂) distributed random variable is given by

  

  for real x ≥ 0, where d₁ and d₂ are positive integers, and B is the beta function.

  The cumulative distribution function is 

  where I is the regularized incomplete beta function.

  Fisher-Snedecor

  Probability density function
  Cumulative distribution function
  parameters:	0,\ d_2>0" src="http://upload.wikimedia.org/math/6/c/3/6c3df1080d0ee2d5084dd8485a1e7ca8.png" style="border-top-style: none; border-right-style: none; border-bottom-style: none; border-left-style: none; border-width: initial; border-color: initial; vertical-align: middle; "> deg. of freedom
  support:
  pdf:
  cdf:
  mean:	for d2 > 2
  median:
  mode:	for d1 > 2
  variance:	for d2 > 4
  skewness:	for d2 > 6