Define Moment Generating Functions
Discuss Properties
The \(k\)th moment is defined as the expectation of the random variable, raised to the \(k\)th power, defined 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:
\[ E(X^k)=\frac{d^km(t)}{dt}\Bigg|_{t=0} \]
Let \(X\) follow a distribution \(f\), with the an MGF \(M_X(t)\), the MGF of \(Y=aX+b\) is given as
\[ M_Y(t) = e^{tb}M_X(at) \]
Let \(X\) and \(Y\) be two random variables with MGFs \(M_X(t)\) and \(M_Y(t)\), respectively, and are independent. The MGF of \(U=X-Y\)
\[ M_U(t) = M_X(t)M_Y(-t) \]
Let \(X\) and \(Y\) have the following distributions \(F_X(x)\) and \(F_Y(y)\) and MGFs \(M_X(t)\) and \(M_Y(t)\), respectively. \(X\) and \(Y\) have the same distribution \(F_X(x)=F_Y(y)\) if and only if \(M_X(t)=M_Y(t)\).
Let \(X_1,\cdots, X_n\) be independent random variables, where \(X_i\sim N(\mu_i, \sigma^2_i)\), with \(M_{X_i}(t)=\exp\{\mu_i t+\sigma^2_it^2/2\}\) for \(i=1,\cdots, n\). Find the MGF of \(Y=a_1X_1+\cdots+a_nX_n\), where \(a_1, \cdots, a_n\) are constants.