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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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The expected value of a function of a RV X is given by:
Joint Characterization of RVs
Jointly Continuous RVs
Two jointly (absolutely) continuous RVs X and Y are defined by a joint PDF:
and a joint CDF:
Jointly continuous RVs have the following properties:
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Examples of Mean, Variance, Standard Deviation, etc.
Continuous RVs
Consider the RV X with uniform distribution over [0,1]:
Suppose h(x) = x3/2:
Discrete RVs
Consider the RV X with binomial distribution and parameters N = 2 and p = 0.2. Support of X is {x0 = 0, x1 = 1, x2 = 2}.
Suppose h(x) = x1/2:
Joint Characterization of RVs
Jointly Continuous RVs
Two jointly (absolutely) continuous RVs X and Y are defined by a joint PDF:
and a joint CDF:
Jointly continuous RVs have the following properties:
Example
Consider RVs X and Y with a joint PDF:
Find Pr(X - Y < 1):
Jointly Discrete RVs
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:
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:
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.
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:
