ch02 JWBK035-Vince February 12, 2007 6:50 Char Count= 0
88 HANDBOOK OF PORTFOLIO MATHEMATICS
where: K=The chi-square variable X^2.
V=The number of degrees of freedom, which is the
single input parameter.
EXP( )=The exponential function.
GAM( )=The standard gamma function.A few notes on the gamma function are in order. This function has the
following properties:1.GAM(O)= 1
2.GAM(1/2)=The square root of pi, or 1.772453851
3.GAM(N) = (N – 1)*GAM(N – 1); therefore, if N is an integer,
GAM(N)=(N−1)!Notice in Equation (2.51) that the only input parameter is V, the num-
ber of degrees of freedom. Suppose that rather than just taking one in-
dependent random variable squared (K^2 ), we take M independent random
variables squared, and take their sum:JM=K^21 +K^22 ...K^2 MNow JMis said to have the Chi-Square Distribution with M degrees of
freedom. It is the number of degrees of freedom that determines the shape
of a particular Chi-Square Distribution. When there is one degree of free-
dom, the distribution is severely asymmetric and resembles the Exponen-
tial Distribution (with A=1). At two degrees of freedom the distribution
begins to look like a straight line going down and to the right, with just a
slight concavity to it. At three degrees of freedom, a convexity starts taking
shape and we begin to have a unimodal-shaped distribution. As the number
of degrees of freedom increases, the density function gradually becomes
more and more symmetric. As the number of degrees of freedom becomes
very large, the Chi-Square Distribution begins to resemble the Normal Dis-
tribution per the Central Limit Theorem.The Chi-Square “Test”
Do not confuse the Chi-Square “Test” with the Chi-Square Distribution. The
former is a hypothesis testing procedure (one of many such procedures).
Mention is made of it here, but it should not be confused with the distribu-
tional form of the same name.
There exist a number of statistical tests designed to determine if two
samples come from the same population. Essentially, we want to know if