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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 X1, with mean u1and standard deviation s1, and random variable X2, with mean u2 and standard deviation s2. Let the correlation coefficient of X1 and X2 be rho. The bivariate normal density is defined as:

f(x1, x2) = exp[(-0.5/(1-rho^2)) * ((x1 - u1)/ s1 )^2) - 2*rho*((x1 - u1)/ s1)*((x2 - u2)/ s2) + ((x2 - u2)/ s2 )^2)]  / [2* pi * s1 * s2 * sqrt(1-rho^2)]

where u1 and s1 are the mean and standard deviation of the marginal distribution of x1:

f1(x1) = exp[-0.5) * ((x1 - u1)/ s1 )^2)]/ [sqrt(2* pi) * s1]

and, where u2 and s2 are the mean and standard deviation of the marginal distribution of x2:

f2(x2) = exp[-0.5) * ((x2 - u2)/ s2 )^2)]/ [sqrt(2* pi) * s2]

Bivariate Normal Density
with u1 = 0, s1 = 1, u2 = 0, s2 = 1, rho = .7

B. Conditional Bivariate Normal Distribution:

The conditional density of x1 given x2

f(x1 | x2) = f(x1, x2) / f2(x2)

f(x1 | x2) = exp[(-0.5/(1-rho^2)) * {(1/s1)*(x1 - (u1 + rho*(s1/s2)*(x2 - u2))}^2] / [sqrt(2* pi) * s1 * sqrt(1-rho^2)]

The mean of f(x1 | x2) is   (u1 + rho*(s1/s2)*(x2 - u2))

while the standard deviation f(x1 | x2) is   s1*sqrt(1-rho^2).

As x2 increases, the graph of f(x1 | x2) 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(x1 | x2) increases in height, and its width decreases about the mean of   f(x1 | x2).

Conditional Bivariate Normal Density
with u1 = 5, s1 = 1.5, u2 = 10, s2 = 2, rho = .7

 
   

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