integrating a bivariate normal

• To: mathgroup at smc.vnet.net
• Subject: [mg13704] integrating a bivariate normal
• From: Dennis Swaney <dennis at shrubbery.com>
• Date: Wed, 19 Aug 1998 01:38:02 -0400
• Sender: owner-wri-mathgroup at wolfram.com

```Dear Mathgroup-

Is it me, or is Mathematica incapable of integrating the bivariate
normal distribution if the variables are correlated?

First, I tried the following (integrating the pdf for uncorrelated
variables):

Integrate[Exp[-1/2*(((x - m1)/s1 )^2 +
((y -  m2)/s2)^2)]/(2*Pi*Sqrt[s1^2]* Sqrt[s2^2]),
{x, -Infinity, Infinity}, {y, -Infinity, Infinity}]

and it produced the correct answer (1).  Next, I tried integrating the
pdf for nonzero r:

Integrate[Exp[-1/(2(1-r^2))*(((x - m1)/s1 )^2 +
((y -  m2)/s2)^2-2 r((x - m1)/s1 ) ((y -  m2)/s2 ))]/(
2*Pi*Sqrt[s1^2]* Sqrt[s2^2]*Sqrt[1-r^2]),
{x, -Infinity, Infinity}, {y, -Infinity, Infinity}]

and it produced a long, erroneous result (a function of x, which is a
variable of integration!)

Is it possible that the problem results because Abs(r) should be < 1?
If so, how can I specify this condition?

thanks

-Dennis

Dennis P. Swaney

email: Dennis at shrubbery.com

```

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