MaplePrimes - answers and comments on Question, Derivative of a Bi-variate Normal distribution

Only by fsolve

Markiyan Hirnyk — Sun, 04 Sep 2011 15:15:32 Z

One doesn't see any solution for Q in a closed form. In view of it the fsolve command should be applied.

See modi.mw and modi.pdf

Concavity

Sobhan — Sun, 04 Sep 2011 16:11:23 Z

thank you Markiyan for your prompt answer.

My other question is how can we check the concavity of a function in Maple both mathemativcally and graphically?

one step further

Axel Vogt — Sun, 04 Sep 2011 16:51:07 Z

functionmodel; diff(%,Q);
convert(%, erfc);
simplify(%, size); collect(%, erfc);

shows it is a cumulative + complementary normal pdf with some
parameters. In some cases it will have no zero at all, mu2 =0.

The behaviour depends on the parameters

Re:concavity

Markiyan Hirnyk — Sun, 04 Sep 2011 20:09:07 Z

@Sobhan

> eval(diff(functionmodel,Q$ 2), [mu[1] = 1, mu[2] = -2, sigma[1] = 4, sigma[2] = 2, rho = .9, Q = 2.835380684]);
                                                       (1/2)
                              0.1320958824 2
                            - -------------------
                                      (1/2)
                                    Pi

but for mu2 = 0 ...

Axel Vogt — Sun, 04 Sep 2011 23:52:24 Z

diff(functionmodel,Q$2): # http://en.wikipedia.org/wiki/Concave_function
simplify(%, size);       # to show the sign 
eval(%, [mu[1] = 1, mu[2] = -2*0, sigma[1] = 4, sigma[2] = 2, rho = .9, Q = 2.835380684]):
evalf(%);

                            0.07414311113

It depends on the parameters, (sigma[1]*mu[2]+sigma[2]*rho*(Q-mu[1])) gives the sign