Egwald Statistics: Probability and Stochastic Processes
by
Elmer G. Wiens
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Distributions  Sampling Distributions  Sample Mean  Hypotheses Testing  Random Walk
 References
Sampling Distributions.
A. ChiSquare Distribution
The chisquare distribution is the gamma distribution with alfa = .5 and beta = nu / 2, where nu = degrees of freedom. If the random variable X has the N(0, 1) distribution, then X^^{2} has the chisquare distribution with nu = 1 degrees of freedom. If each random variable of { X_{1}, ..., X_{n}} has the N(0,1) distribution, and
Y = X_{1}^^{2} + . . . + X_{n}^^{2}
then the random variable Y has the chisquare distribution with nu = n degrees of freedom.
The density of the chisquare distribution is:
f(y) = K * .5^^{nu/2} * y^^{(nu2)/2} * exp(y/2)
where K = 1 / G(nu/2) and G = the gamma function.
Density of the ChiSquare Distribution with nu degrees of freedom
B. Studentt Distribution
The density of the Studentt distribution is:
f(t) = G(nu/2 + 1/2) * (1 + t^^{2} / nu)^^{(nu+1)/2} / sqrt(nu * pi) * G(nu/2)
where G is the gamma function and nu = degrees of freedom.
Density of the Studentt Distribution
If the random variable X has the standard normal distribution, and the random variable Y has the chisquare distribution with nu degrees of freedom, then the random variable T defined as:
T = X / sqrt(Y/nu)
has the Studentt distribution with nu degrees of freedom.
C. F Distribution
The density of the F distribution with r and s degrees of freedom is:
f(x) = c * x^^{(r/s)1} / (1 + r * x / s)^^{(r + s)/2}
where c = G((r + s)/2) * (r/s)^^{(r/2)} / [G(r/2) * G(s/2)] and G is the gamma function.
Density of the F Distribution
If the independent random variables U and V have chisquare distributions with r and s degrees of freedom, respectively, then the random variable
F = (U / r) / (V / s)
has the F distribution with r and s degrees of freedom.
