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Integer Series Identities
If 0 < p < 1, then
If a != ≠ 1, then
If a != ≠ 1, then
where, k factorial = k! = (k)(k - 1)(k - 2)(...)(1).
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Two jointly discrete RVs X and Y are characterized by a support:
a joint PMF {pi,j} for all i and j, and a joint CDF given by:
with the following properties:
Marginal PDFs
The PDFs of the individual absolutely continuous RVs, or marginal PDFs, are determined from the joint PDF by:
Marginal PMFs
The PMFs of the individual absolutely discrete RVs, or marginal PMFs, are determined from the joint PMF by:
Shorthand Notation
If no ambiguity results, subscripts or superscripts for the PDF, PMF, or CDF are often omitted. For example:
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:
If fY(y) ≠ 0:
So, fX|Y(y) is a valid PDF for the RV X given Y = y. This is written as X | Y = y.
Conditional CDF
If X and Y are jointly absolutely continuous RVs and fY(y) ≠ 0, the conditional CDF of X given Y = y is defined as:
If X and Y are jointly absolutely continuous RVs and Pr(Y in the set B) ≠ 0, the conditional CDF of X given Y in the set B is defined as:
Examples
Example 1
B = (-∞, a], then Y in the set B ↔ Y ≤ a.
Example 2
B = [0,1] U [3,4], then Y in the set B ↔ (0 ≤ Y ≤ 1 or 3 ≤ Y ≤ 4).
Example 3
For 0 ≤ y ≤ 1:
For y < 0 or y > 1,
fX|Y(x|y) = 0 and FX|Y(x|y) is not defined.