Moment Generating Function

Results

The moment generating function of a real-valued random variable is the function mapping real numbers to the expectation of the exponential of the product of that real number with the random variable.

Notation

Let $(X, \mathcal{A} , \mu )$ be a probability space. Let $f$ be a real-valued random variable on $X$. Let $R$ denote the real numbers. For each $t \in R$, denote by $tf$ the function $x \mapsto tf(x)$. Similarly, denote by $e^{tf}$ the funciton $x \mapsto e^{tf(x)}$.

Denote the moment generating function of $f$ by $m_f: R \to R$. We defined it by

\[ m_f(t) = \E (e^{tf}). \]