Moment Generating Functions (MGFs) are a powerful tool in probability and statistics, particularly useful for deriving moments of a random variable and proving the uniqueness of probability distributions. They are especially relevant in actuarial exams for their ability to simplify complex calculations and establish key properties of distributions.

The Moment Generating Function (MGF) of a random variable $X$, denoted by $M_X(t)$, is defined as the expected value of $e^{tX}$, provided that this expectation exists for all $t$ in some open interval containing 0. Mathematically, it is expressed as:

$M_X(t) = E[e^{tX}] = \int_{-\infty}^{\infty} e^{tx} f(x) dx$ (for continuous random variables) $M_X(t) = E[e^{tX}] = \sum_{x} e^{tx} P(X=x)$ (for discrete random variables)

The relationship between the MGF and the moments of a random variable is a cornerstone of its utility. By expanding the $e^{tX}$ term in its Taylor series around $t=0$, we get: $e^{tX} = 1 + tX + \frac{(tX)^2}{2!} + \frac{(tX)^3}{3!} + \dots$ Taking the expected value of both sides: $M_X(t) = E[e^{tX}] = E\left[1 + tX + \frac{t^2X^2}{2!} + \frac{t^3X^3}{3!} + \dots\right]$ By linearity of expectation: $M_X(t) = 1 + tE[X] + \frac{t^2E[X^2]}{2!} + \frac{t^3E[X^3]}{3!} + \dots$ Comparing this to the Taylor series expansion of $M_X(t)$ around $t=0$, which is $M_X(t) = M_X(0) + M_X'(0)t + \frac{M_X''(0)}{2!}t^2 + \frac{M_X'''(0)}{3!}t^3 + \dots$, we can equate the coefficients of the powers of $t$ to find the moments:

• $M_X(0) = E[e^{0X}] = E[1] = 1$
• $M_X'(0) = E[X]$ (The first moment about the origin, i.e., the mean)
• $M_X''(0) = E[X^2]$ (The second moment about the origin)
• $M_X^{(k)}(0) = E[X^k]$ (The $k$-th moment about the origin)

A fundamental theorem states that if two random variables $X$ and $Y$ have MGFs $M_X(t)$ and $M_Y(t)$ respectively, and these MGFs exist in an open interval containing 0, then $X$ and $Y$ have the same distribution if and only if $M_X(t) = M_Y(t)$ for all $t$ in that interval. This property is invaluable for identifying distributions, especially when dealing with sums of independent random variables.

It's important to note that not all random variables have an MGF that exists for all $t$ in an open interval around 0. For instance, the Cauchy distribution does not have an MGF. In such cases, alternative tools like the Characteristic Function (which always exists) are used.

A significant application of MGFs is in determining the distribution of the sum of independent random variables. If $X_1, X_2, \dots, X_n$ are independent random variables with MGFs $M_{X_1}(t), M_{X_2}(t), \dots, M_{X_n}(t)$ respectively, then the MGF of their sum $S = X_1 + X_2 + \dots + X_n$ is the product of their individual MGFs: $M_S(t) = M_{X_1}(t) M_{X_2}(t) \dots M_{X_n}(t)$ This property, combined with the uniqueness theorem, allows us to identify the distribution of sums of independent random variables, which is a frequent topic in actuarial exams.