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. 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, ..., X_n} has the N(0,1) distribution, and

Y = X₁^² + . . . + X_n^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)

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}

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.