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If no ambiguity results, subscripts or superscripts for the PDF, PMF, or CDF are often omitted. For example:
Cross-Correlation
Continuous RVs
The cross-correlation of two jointly absolutely continuous RVs X and Y is given by:
Discrete RVs
The cross-correlation of two jointly discrete RVs X and Y is given by:
Covariance
The covariance of two RVs X and Y is given by:
If the Cov(X,Y) = 0, X and Y are uncorrelated.
Independence
Continuous RVs
Two jointly absolutely continuous RVs X and Y are independent if:
Or equivalently:
Discrete RVs
Two jointly discrete RVs X and Y are independent if:
Or equivalently:
Conditional Probability
Continuous RVs
Conditional PDF
If X and Y are jointly absolutely continuous RVs, the conditional PDF of X given Y is defined as:
...
For y < 0 or y > 1, fX|Y(x|y) = 0 and FX|Y(x|y) is not defined.
Discrete RVs
Conditional PMF
Suppose X and Y are jointly discrete RVs with support {(xi,yj)}, i and j integers. The conditional PMF of X given Y is defined as:
If pjY != 0:
So, pi|jX|Y, i = ..., -1, 0, 1, ... is a valid PMF for the RV X given Y = yj, with support {..., x-1, x0, x1, ...}.
Conditional CDF
If X and Y are jointly discrete RVs and pjY != 0, the conditional CDF of X given Y = yj is defined as:
If X and Y are jointly discrete RVs and Pr(Y in the set B) != 0, the conditional CDF of X given Y in the set B is defined as:
