Co-Variance and Signum Function

[Hanspi]

I have a problem that turned up in my research. I'm a microelectronics engineer, s I hope this is not a textbook question for you physics specialists :-)

Given a random variable X that produces real numbers x with a distribution p(x).

The random variable Y is generated from X by the signum function; i.e., y=1 for x>=0 and y=-1 for x<0.

How can I calculate the covariance of X and Y in general? And, if there is no general solution, does a solution exist if p(x) is a gaussian distribution with mean zero?

Slainte!
Hanspeter

 

[Stephen Tashi]

If we use E(X) to denote the expected value of X, you need to compute
E(XY) - E(X)E(Y). The expectations would be computed as integrals. For example, to compute E(XY)

$$E(XY) = \int_0^\infty x(1) p(x) dx + \int_{-\infty}^0 x(-1) p(x) dx$$
$$= \int_0^\infty x p(x)dx - \int_{-\infty}^0 x p(x) dx$$

 

[Hanspi]

Thanks a lot for the reply, this is indeed what I needed. If E(X)=0, then E(Y)=0 and the Covariance is simply the mean value of the positive half of p(x) minus the mean value of the negative half of p(x).

To test my understanding, I did (MATLAB):

c=randn(1,1000000);
d=sign(c);
cov(c,d)

ans =

1.0000 0.7981
0.7981 1.0000

mean(c.*(d>=0))-mean(c.*(d<0))

ans =

0.7981

Now comes the mathematically more difficult part: to calculate p(x) in my system depending on the parameters of the system. But hat I know how to to, so thanks a lot for giving me such a good start.

Hanspeter