Expected Values
Variances
Properties of Expected Values
The expected value is the value we expect when we randomly sample from population that follows a specific distribution. The expected value of \(Y\) is
\[ E(Y)=\sum_y yP(y) \]
where \(P(y)\) is the PMF of \(Y\).
The expected value of a function of a random variable \(Y\) is provided as
\[ E\{g(Y)\} = \sum_y g(y)P(y) \]
The variance is the expected squared difference between the random variable and expected value.
\[ Var(Y)= E[\{Y-E(Y)\}^2]=\sum_y\{y-E(Y)\}^2P(y) \]
\[ Var(Y) = E(Y^2) - E(Y)^2 \]