Moments
Moment Generating Functions
Properties
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 \(Y\).
Moments are numerical measures that describe the shape and characteristics of a random variable’s probability distribution.
The \(k\)th moment is described as
\[ E(X^k) \]
The moment generating functions is used to obtain the \(k\)th moment. The mgf is defined as
\[ m(t) = E(e^{tX}) \]
The \(k\)th moment can be obtained by taking the \(k\)th derivative of the mgf, with respect to \(t\), and setting \(t\) equal to 0:
\[ E(X^k)=\frac{d^km(t)}{dt^k}\Bigg|_{c=0} \]