Properties of the Distribution

Calculation of Conditional Mean and Variance

Because we are dealing with a joint distribution of two variables, we will consider the conditional means and variances of X and Y for fixed y and x, respectively. The means and variances of the marginal distributions were given in the first section of the worksheet. Interestingly, the conditional densities of X and Y are normal distributions as well. This can be shown easily by examining the conditional densities.

> restart:

> with(plots,display,textplot3d):

> f(x,y):=exp((-1/(2*(1-rho^2)))*(((x-mu1)/sigma1)^2-2*rho*(x-mu1)*(y-mu2)/(sigma1*sigma2)+((y-mu2)/sigma2)^2))/(2*Pi*sigma1*sigma2*sqrt(1-rho^2)):

> assume(sigma1>0):assume(sigma2>0):

> g(x):=int(f(x,y),y=-infinity..infinity):

So, the conditional density of Y given X = x is

> f[givenX](y):=simplify((f(x,y)/g(x)));

and the conditional expectation of Y given X = x is

> EY[givenX]:=simplify(int(y*f[givenX](y),y=-infinity..infinity));

> E_Y_SQ[givenX]:=int((y^2)*f[givenX](y),y=-infinity..infinity);

Calculating the conditional variance using the typical computational formula:

> VarY[givenX]:=E_Y_SQ[givenX]-EY[givenX]^2;

Similarly, the conditional mean and variance for X given Y = y are and .

Moment Generating Function for the Bivariate Normal Distribution

The joint moment generating function for two random variables X and Y is given by .

We now find this MGF for the bivariate normal distribution.

> restart:

> with(plots,display,textplot3d): with(student):

> f(x,y):=exp((-1/(2*(1-rho^2)))*(((x-mu1)/sigma1)^2-2*rho*(x-mu1)*(y-mu2)/(sigma1*sigma2)+((y-mu2)/sigma2)^2))/(2*Pi*sigma1*sigma2*sqrt(1-rho^2)):

> assume(sigma1>0):assume(sigma2>0):

> value(Doubleint(f(x,y)*exp(t[1]*x+t[2]*y),x=-infinity..infinity,y=-infinity..infinity));

So, the MGF of a bivariate normal distribution is given by

We now define this MGF as a function of and .

> M[X,Y](t[1],t[2]):=exp(t[1]*mu[1]+t[2]*mu[2]+1*(sigma[1]^2*t[1]^2+2*rho*sigma[1]*sigma[2]*t[1]*t[2]+sigma[2]^2*t[2]^2)/2);

The joint MGF provides us with alternative ways of finding the means of the marginal distributions as well as an alternative method of finding the mean and variance of the marginal distributions as well as an alternative method of finding Cov( X, Y ) by way of the following formulas:

Let's start by finding the mean of the marginal distribution of X :

> EX:=simplify(subs(t[1]=0,t[2]=0,diff(M[X,Y](t[1],t[2]),t[1])));

which is what we expected. Now we use the computational formula for variance to find the variance of the marginal distribution of X :

> E_X_SQ:=simplify(subs(t[1]=0,t[2]=0,diff(M[X,Y](t[1],t[2]),t[1]\$2)));

> VarX:=E_X_SQ-EX^2;

which is also what we expected. Similarly, we will use the computational formula for covariance to find Cov( X , Y ):

> E_XY:=simplify(subs(t[1]=0,t[2]=0,diff(M[X,Y](t[1],t[2]),t[1],t[2])));

> EY:=simplify(subs(t[1]=0,t[2]=0,diff(M[X,Y](t[1],t[2]),t[2])));

> CovXY:=E_XY-EX*EY;