PROBABILITY 09
Expectation
Expectation is the probability-weighted average of a random variable and describes its mean value across repeated realisations.
Discrete expectation
For a discrete random variable,
1. Continuous expectation
For a continuous variable with density \(f_X\),
when the integral exists.
2. Expectation is not necessarily a possible outcome
A fair die has
\[E[X]=3.5,\]although \(3.5\) cannot occur on a single roll. Expectation describes the distribution, not necessarily one realisation.
3. Functions of random variables
For a suitable function \(g\),
\[E[g(X)]=\sum_x g(x)P(X=x)\]in the discrete case, or
\[E[g(X)]=\int g(x)f_X(x)\,dx\]in the continuous case.
4. Linearity
Linearity does not require \(X\) and \(Y\) to be independent.
5. Products
If \(X\) and \(Y\) are independent and expectations exist,
\[E[XY]=E[X]E[Y].\]This is not true in general without independence.
6. Expectation in stochastic modelling
If \(X(t)\) is a stochastic process, then \(E[X(t)]\) describes its mean state at each time. It does not describe the uncertainty around that mean.