Fall 2011 Strategic Practice 6: Section 2 (Moment Generating Functions MGFs) - Question 4
Let , 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 (the sum starts at j = 1 because g(0) = 0), the coefficient is called the jth cumulant of X. Find the jth cumulant of X, for all .
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, butstatistics is the logic of uncertainty."