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Egwald Statistics: Probability and Stochastic Processes


Elmer G. Wiens

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

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!)]


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