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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:

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and a joint CDF:

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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) = x^3/2:

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) = x^1/2:

Joint Characterization of RVs

Jointly Continuous RVs

Two jointly (absolutely) continuous RVs X and Y are defined by a joint PDF:

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and a joint CDF:

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Jointly continuous RVs have the following properties:

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Two jointly discrete RVs X and Y are characterized by a support:

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