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:
Follow Elmer Wiens on Twitter:
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; ø^{2} = variance; ø = standard deviation. Then the density of the normal distribution is:
f(x) = exp[0.5 * ((x  µ)/ ø )^^{2})] / (2* pi * ø^{2})^^{.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^^{(beta1)} * 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!)]
