Fall 2011 Strategic Practice 6: Section 2 (Moment Generating Functions MGFs) - Question 4
Let $X\sim Expo(\lambda )$, and let M(t) be the MGF of X. The cumulant generating function is defined to be g(t) = ln M(t). Expanding g(t) as a Taylor series $g(t)=\sum_{j=1}^{\infty }\frac{c_{j}}{j!}t^{j}$ (the sum starts at j = 1 because g(0) = 0), the coefficient $c_{j}$ is called the jth cumulant of X. Find the jth cumulant of X, for all $j\geq 1$.
Solution: Using the Taylor series for e^t, we find that the jth cumulant is lambda for all j greater than or equal to 1.
"Mathematics is the logic of certainty, but statistics is the logic of uncertainty."
Copyright © 2011 Stat 110 Harvard. Website layout by former Stat110'er.