www.egwald.com Egwald Web Services

Egwald Web Services
Domain Names
Web Site Design

Egwald Website Search JOIN US AS A FACEBOOK FAN Twitter - Follow Elmer WiensRadio Podcasts - Geraldos Hour

 

Statistics Programs - Econometrics and Probability Economics - Microeconomics & Macroeconomics Operations Research - Linear Programming and Game Theory Egwald's Mathematics Egwald's Optimal Control
Egwald HomeStatistics Home PageMultiple RegressionProbability and Stochastic Processes
 

Egwald Statistics: Probability and Stochastic Processes

by

Elmer G. Wiens

Egwald's popular web pages are provided without cost to users.
Please show your support by joining Egwald Web Services as a Facebook Fan: JOIN US AS A FACEBOOK FAN
Follow Elmer Wiens on Twitter: Twitter - Follow Elmer Wiens

Distributions | Sampling Distributions | Sample Mean | Hypotheses Testing | Random Walk | References

Sampling Distributions.

A. Chi-Square Distribution

The chi-square 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 chi-square distribution with nu = 1 degrees of freedom. If each random variable of { X1, ..., Xn} has the N(0,1) distribution, and

Y = X1^2 + . . . + Xn^2

then the random variable Y has the chi-square distribution with nu = n degrees of freedom.

The density of the chi-square distribution is:

f(y) = K * .5^nu/2 * y^(nu-2)/2 * exp(-y/2)

where K = 1 / G(nu/2) and G = the gamma function.

Density of the Chi-Square Distribution with nu degrees of freedom

B. Student-t Distribution

The density of the Student-t 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 Student-t Distribution

If the random variable X has the standard normal distribution, and the random variable Y has the chi-square distribution with nu degrees of freedom, then the random variable T defined as:

T = X / sqrt(Y/nu)

has the Student-t 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 chi-square 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.

 
   

      Copyright © Elmer G. Wiens:   Egwald Web Services       All Rights Reserved.    Inquiries