The moments of a distribution provide key insights into its shape and properties. They quantify aspects such as the mean, variance, and skewness.
The \(n\)-th moment of a probability distribution is the expected value of the \(n\)-th power of deviations from a fixed point, typically the mean. However, when dealing with a moment generating function (mgf) like in this problem, the moments can be extracted directly from the series expansion.
Thus, for the function \(M(t) = (1-t)^{-3}\), its Maclaurin series expansion gives us the coefficients directly as the moments:
- First moment (Mean): coefficient of \(t\), which is \(-3\).
- Second moment: coefficient of \(t^2\), which is \(6\).
Recognizing these coefficients helps in understanding the behavior and characteristics of the underlying distribution.