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