Egwald Statistics: Probability and Stochastic Processes
by
Elmer G. Wiens
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Distributions  Sampling Distributions  Sample Mean  Hypotheses Testing  Random Walk
 References
Probability Distributions.
Univariate Normal  Bivariate Normal
A. Bivariate Normal Distribution:
Given the random variable X_{1}, with mean u_{1}and standard deviation s_{1}, and random variable X_{2}, with mean u_{2} and standard deviation s_{2}. Let the correlation coefficient of X_{1} and X_{2} be rho. The bivariate normal density is defined as:
f(x_{1}, x_{2}) = exp[(0.5/(1rho^^{2})) * ((x_{1}  u_{1})/ s_{1} )^^{2})  2*rho*((x_{1}  u_{1})/ s_{1})*((x_{2}  u_{2})/ s_{2}) + ((x_{2}  u_{2})/ s_{2} )^^{2})] / [2* pi * s_{1} * s_{2} * sqrt(1rho^^{2})]
where u_{1} and s_{1} are the mean and standard deviation of the marginal distribution of x_{1}:
f_{1}(x_{1}) = exp[0.5) * ((x_{1}  u_{1})/ s_{1} )^^{2})]/ [sqrt(2* pi) * s_{1}]
and, where u_{2} and s_{2} are the mean and standard deviation of the marginal distribution of x_{2}:
f_{2}(x_{2}) = exp[0.5) * ((x_{2}  u_{2})/ s_{2} )^^{2})]/ [sqrt(2* pi) * s_{2}]
Bivariate Normal Density with u1 = 0, s1 = 1, u2 = 0, s2 = 1, rho = .7
B. Conditional Bivariate Normal Distribution:
The conditional density of x_{1} given x_{2}
f(x_{1}  x_{2}) = f(x_{1}, x_{2}) / f_{2}(x_{2})
f(x_{1}  x_{2}) = exp[(0.5/(1rho^^{2})) * {(1/s_{1})*(x_{1}  (u_{1} + rho*(s_{1}/s_{2})*(x_{2}  u_{2}))}^^{2}] / [sqrt(2* pi) * s_{1} * sqrt(1rho^^{2})]
The mean of f(x_{1}  x_{2}) is (u_{1} + rho*(s_{1}/s_{2})*(x_{2}  u_{2}))
while the standard deviation f(x_{1}  x_{2}) is s_{1}*sqrt(1rho^^{2}).
As x_{2} increases, the graph of f(x_{1}  x_{2}) shifts to the right.
Conditional Bivariate Normal Density with u1 = 5, s1 = 1.5, u2 = 10, s2 = 2, rho = .3
As rho increases, from 0 to 1, the graph of f(x_{1}  x_{2}) increases in height, and its width decreases about the mean of f(x_{1}  x_{2}).
Conditional Bivariate Normal Density with u1 = 5, s1 = 1.5, u2 = 10, s2 = 2, rho = .7
