Expectation

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  1. 1. Definition
  2. 2. Example

1.  Definition

Consider a random variable \fs4 X on the probability space \fs4 (\Omega, \cal{F}, P). Then, the expectation of \fs4 X, denoted with \fs4 EX, is given by

\fs4 EX = \int_\Omega X(\omega) dP(\omega)

If the random variable is discrete, then equivalently we can write

\fs4 EX = \sum_{\omega \in \Omega} X(\omega) P(\omega)

2.  Example

We assume here the setting given in the example of a random variable.

Say that we are interested in the expectation \fs4 EX on \fs4 (\Omega, \mathscr{F}, P). Since \fs4 X is \fs4 \mathscr{F}-measurable, the expectation is valid and we can apply the definition

\fs4 EX = \sum_{\omega \in \Omega} X(\omega) P(\omega)

If we also assume that the coin is fair when determining the probability measure \fs4 P, then all \fs4 \omega \in \Omega will have equal probabilities \fs4 P(\omega) = 1/4. Therefore

\fs4 EX = 1/4 \cdot (X(HH) + X(HT) + X(TH) + X(TT)) = 1/4 \cdot (2+1+1+0) = 1

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