# Processes: Orthogonal, Uncorrelated, Statistically Independent

How are they all related? You can define them as:

• Orthogonal Processes: $$E[XY] = 0$$
• Uncorrelated Processes: $$E[XY] = E[(X - \mu_x)(Y - \mu_y)] = 0$$
• Statistically Independent Processes: $$E[XY] = E[X] \cdot E[Y]$$

If two processes are orthogonal:

• they are also uncorrelated
• they are not necessary independent

If two processes are uncorrelated:

• they are not necessary orthogonal
• they are not necessary independent

If two processes are independent:

• they are uncorrelated
• they are orthogonal

Is that correct? I'm not sure.

• – MaxFrost May 18 at 11:31
• What is the relevance of these questions to random processes? – Dilip Sarwate May 18 at 16:11

You got some definitions wrong. It's correct that orthogonality means that $$E[XY]=0$$. Uncorrelated means that $$X-\mu_X$$ and $$Y-\mu_Y$$ are orthogonal, i.e., $$E[(X-\mu_X)(Y-\mu_Y)]=0$$. If you work that out you should arrive at the equivalent condition $$E[XY]=\mu_X\mu_Y$$ for uncorrelatedness (not for independence!). Consequently, if at least one of the two RVs $$X$$ and $$Y$$ have a zero mean, then orthogonality implies uncorrelatedness and vice versa.

Statistical independence means that the joint PDF of two random variables can be written as the product of the individual PDFs:

$$f_{XY}(x,y)=f_X(x)f_Y(y)\tag{1}$$

Independence implies uncorrelatedness, but the opposite is generally not true. If $$X$$ and $$Y$$ are jointly Gaussian, then independence and uncorrelatedness are equivalent. Consequently, in the special case that $$X$$ and $$Y$$ are jointly Gaussian and at least one of them has a mean of zero, then orthogonality, uncorrelatedness, and independence are all equivalent.

• So independence does not imply orthogonality? – user674907 May 18 at 12:21
• @user674907: No, unless one of the two RVs has a zero mean. – Matt L. May 18 at 12:51