0
$\begingroup$

The probability density function of binary normal distribution in textbooks is in the following form:

enter image description here

Refine[PDF[
  MultinormalDistribution[{μ1, μ2}, {{σ1^2, ρ*\
σ1*σ2}, {ρ*σ1*σ2, σ2^2}}], {x,
    y}], σ1 > 0 && σ2 > 0 && ρ > 0]
(-(1/(2 (1 - ρ^2) )) ((x - μ1)^2/ (σ1^2)  - (
     2 ρ (x - μ1) (y - μ2))/( σ1 σ2) + (y \
- μ2)^2/ (σ2^2) ) // Expand)

(1/2 (-(((x - μ1) (μ1 σ2 - μ2 ρ σ1 - \
σ2 x + ρ σ1 y))/((ρ^2 - 
         1) σ1^2 σ2)) - ((y - μ2) (-(μ1 ρ \
σ2) + μ2 σ1 + ρ σ2 x + σ1 \
(-y)))/((ρ^2 - 1) σ1 σ2^2)) // Expand)

Although the expansion form of PDF[MultinormalDistribution[{μ1, μ2}, {{σ1^2, ρ* σ1*σ2}, {ρ*σ1*σ2, σ2^2}}], {x, y}] is consistent with the textbook, I still want to reduce it to the following output form and keep it unchanged:

1/(2 π*σ1*σ2*(1 - ρ^2)^(1/2))*
 E^(-(1/(2 (1 - ρ^2))) ((x - μ1)^2/(σ1^2) - (2 ρ \
(x - μ1) (y - μ2))/(σ1 σ2) + (y - μ2)^2/(\
σ2^2)))

$$ \frac{1}{2\pi\sigma1\sigma2\sqrt{(1-\rho^{2})}} \mathrm{e}^{-\frac{1}{2(1-\rho^{2})}(\frac{(x-\mu1)^{2}}{\sigma1^{2}}-\frac{2\rho(x-\mu1)(y-\mu2)}{\sigma1\sigma2}+\frac{(y-\mu2)^{2}}{\sigma2^{2}})} $$

What should I do?

$\endgroup$
3
  • 3
    $\begingroup$ Use BinormalDistribution, i.e., PDF[BinormalDistribution[{μ1, μ2}, {σ1, σ2}, ρ], {x, y}] $\endgroup$ – Bob Hanlon Feb 22 '20 at 2:03
  • 1
    $\begingroup$ @BobHanlon Thank you very much, but how to reduce the above content to the required output format through Simplify and Collect functions, etc. $\endgroup$ – A little mouse on the pampas Feb 22 '20 at 2:09
  • $\begingroup$ A starting point: Simplify[MapAll[Factor, PDF[MultinormalDistribution[{μ1, μ2}, {{σ1^2, ρ*σ1*σ2}, {ρ*σ1*σ2, σ2^2}}], {x, y}]], σ1 > 0 && σ2 > 0 && ρ > 0] /. Power[p_, q_] :> Power[p, Normal[Simplify[Series[Numerator[q], {y, μ2, 2}]]]/Denominator[q]]. $\endgroup$ – J. M.'s ennui Apr 25 '20 at 10:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.