Moment Generating Function Convergence Theorem in Excel

The Moment Generating Function Convergence Theorem is a result in probability theory that relates the convergence of moment generating functions (mgfs) to the convergence of probability distributions. It states that if a sequence of random variables Xn has mgfs Mn(t) that converge pointwise to a function M(t) on an interval around zero, and M(t) is the mgf of another random variable X, then Xn converges in distribution to X. This means that the cumulative distribution functions (cdfs) of Xn approach the cdf of X as n goes to infinity.

The theorem is useful for proving the central limit theorem, which states that the sum of independent and identically distributed random variables with finite variance converges to a normal distribution. The proof relies on showing that the mgfs of the normalized sums converge to the mgf of the normal distribution, and then applying the Moment Generating Function Convergence Theorem to conclude that the distributions also converge.

The theorem can also be used to compute the moments of a random variable, by taking derivatives of the mgf and evaluating them at zero. The n-th moment of X is equal to the n-th derivative of M(t) at zero, if it exists. This follows from the Taylor series expansion of M(t) around zero, which involves the moments of X. However, not all random variables have mgfs, or finite moments, so the theorem does not apply in those cases.

Basic Theory:

The Moment Generating Function (MGF) of a random variable X is defined as:

$M_X(t) = E(e^{tX})$

where $E$ is the expected value. If the MGFs of two random variables X and Y are equal for all values of t within a certain range, then X and Y have the same distribution.

Procedures:

Define the MGFs: Express the MGFs of the random variables you are comparing. Ensure that the formulas are correctly implemented in Excel.
Set the Range: Specify the range of t values for which you want to check convergence. This range should be within the domain of convergence of the MGFs.
Evaluate MGFs: Use Excel formulas to evaluate the MGFs for each random variable over the specified range of t values.
Compare MGFs: Compare the MGFs of the two random variables. If the MGFs are equal within the specified range, the variables are considered to converge in distribution.

Comprehensive Explanation:

Let’s consider a scenario where we have two random variables, X and Y, with the following MGFs:

$M_X(t) = \frac{1}{1-t}$

$M_Y(t) = \frac{2}{2-t}$

We want to determine if X and Y converge in distribution. We’ll set the range of t values as $-1 < t < 1$ for this example.

Excel Implementation:

Create an Excel table with columns for t values, $M_X(t)$ , and $M_Y(t)$ . Fill in the t values and use Excel formulas to calculate the corresponding MGFs.

t	$M_X(t)$	$M_Y(t)$
-0.9	10	-20
-0.5	2	-4
0	#DIV/0! (undefined)	#DIV/0! (undefined)
0.5	2	4
0.9	10	20

Result:

The MGFs are not equal for all values of t within the specified range. Therefore, based on the Moment Generating Function Convergence Theorem, we conclude that X and Y do not converge in distribution.

Other Approaches:

Use Probability Distributions in Excel: Instead of calculating MGFs manually, leverage Excel’s built-in probability distribution functions. For example, you can use the EXPONDIST function for exponential distributions.
Monte Carlo Simulation in Excel: Conduct a Monte Carlo simulation to generate random samples from each distribution and observe the convergence empirically.
Advanced Statistical Tools: Excel also supports more advanced statistical tools, such as the Analysis ToolPak, which provides functions for regression, analysis of variance, and other statistical analyses.