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and for any k ∈ X(Ω),

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 – X or the distribution of X is said to be integrable (or summable) if

 – If X is integrable, then the expectation of X is the real number defined by

EXAMPLE 1.18.– The random variable X defined in Example 1.12 admits an expectation equal to

The following proposition establishes a link between the expectation of a discrete, random variable and measure theory.

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Then,

Let us look at some of the properties of expectations.

PROPOSITION 1.4.– Let X and Y be two integrable, discrete random variables, a, b ∈ ℝ. Then,

1 1) Linearity: [aX + bY ] = a[X]+ b[Y ].

2 2) Transfer theorem: if g is a measurable function such that g(X) is integrable, then

3 3) Monotonicity: if X ≤ Y almost surely (a.s.), then [X] ≤ [Y].

4 4) Cauchy–Schwartz inequality: If X2 and Y2 are integrable, then XY is integrable and

5 5) Jensen inequality: if g is a convex function such that g(X) is integrable, then,

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Variance satisfies the following properties.

PROPOSITION 1.5.– If a discrete random variable X admits variance, then,