12F (Probability)

σ_{1}, σ_{2} > 0

and

∣ ρ ∣ < 1

. Put

a = (x_{1} - μ_{1}) / σ_{1}

and

b = (x_{2} - μ_{2}) / σ_{2}

a^{2} - 2 ρ ab + b^{2} = (b - ρ a)^{2} + (1 - ρ^{2}) a^{2} .

Consequently the joint probability density function factors as

f_{X_{1}, X_{2}} (x_{1}, x_{2}) = \frac{e ^{- a^{2} /2}}{2 π σ _{1}} \frac{e x p [ - ( b - ρ a ) ^{2} / ( 2 ( 1 - ρ ^{2} ))]}{2 π σ _{2} 1 - ρ ^{2}} .

For fixed

x_{1}

, the second factor is a normalized normal distribution density in

x_{2}

, with mean

μ_{2} + ρ σ_{2} a

σ_{2} 1 - ρ^{2}

. Its integral is one; explicitly substitute

w = (b - ρ a) / 1 - ρ^{2}

in the Gaussian integral. Thus the marginal density is

f_{X_{1}} (x_{1}) = \frac{1}{2 π σ _{1}} exp [- \frac{( x _{1} - μ _{1} ) ^{2}}{2 σ _{1}^{2}}] .

b

The other marginal is

X_{2} \sim N (μ_{2}, σ_{2}^{2})

by the same square-completion argument with the indices exchanged. Its moment-generating function is therefore

M_{X_{2}} (t) = E [e^{t X_{2}}] = exp (μ_{2} t + \frac{σ _{2}^{2} t ^{2}}{2}), t \in R .

Now complete the square in the other order:

a^{2} - 2 ρ ab + b^{2} = (a - ρ b)^{2} + (1 - ρ^{2}) b^{2} .

f_{X_{2}} (x_{2}) = \frac{1}{2 π σ _{2}} e^{- b^{2} /2} .

The factor

e^{- b^{2} /2}

f_{X_{1} ∣ X_{2} = x_{2}} (x_{1}) = \frac{1}{2 π σ _{1} 1 - ρ ^{2}} exp [- \frac{( a - ρ b ) ^{2}}{2 ( 1 - ρ ^{2} )}] .

Since

a - ρ b = \frac{x _{1} - ( μ _{1} + ρ σ _{1} ( x _{2} - μ _{2} ) / σ _{2} )}{σ _{1}},

this is a normalized normal distribution density. Hence

X_{1} ∣ X_{2} = x_{2} \sim N (μ_{1} + ρ \frac{σ _{1}}{σ _{2}} (x_{2} - μ_{2}), σ_{1}^{2} (1 - ρ^{2})) .

Conditioning on an exact observation is interpreted through this conditional probability density, rather than conditioning by a ratio of event probabilities.

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