Egwald Statistics — Probability and Stochastic Processes: Normal and Gamma Distributions

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Egwald Statistics: Probability and Stochastic Processes

by

Elmer G. Wiens

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Probability Distributions.

Univariate Normal | Bivariate Normal

A. Normal Distribution: N(µ, ø^²)

Let µ = mean; ø² = variance; ø = standard deviation. Then the density of the normal distribution is:

f(x) = exp[-0.5 * ((x - µ)/ ø )^²)] / (2* pi * ø²)^^.5

Density of the Normal Distribution with µ = 0

When µ = 0 and ø^² = 1, the distribution is called the standard normal distribution, denoted by N(0, 1)

If the random variable X has a N(µ,ø^²⁾ distribution, then the random variable Z = (X - µ)/ø has a N(0,1) distribution.

B. Gamma Distribution

Let the parameters of the gamma distribution be alfa and beta. Then the gamma density is:

f(x) = K * alfa^^beta * x^^(beta-1) * exp(-alfa * x)

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

Density of the Gamma Distribution

C. Gamma Function

The gamma function generally involves an integral. But, if n is an integer then:

G(n) = (n - 1)!

G(n + 1/2) = sqrt(pi) * (2*n)! / [(4^ⁿ) * n!)]