Geometric Distribution - Moments and Cumulants

Moments and Cumulants

The expected value of a geometrically distributed random variable X is 1/p and the variance is (1 − p)/p2:

$\mathrm{E}(X) = \frac{1}{p}, \qquad\mathrm{var}(X) = \frac{1-p}{p^2}.$

Similarly, the expected value of the geometrically distributed random variable Y is (1 − p)/p, and its variance is (1 − p)/p2:

$\mathrm{E}(Y) = \frac{1-p}{p}, \qquad\mathrm{var}(Y) = \frac{1-p}{p^2}.$

Let μ = (1 − p)/p be the expected value of Y. Then the cumulants of the probability distribution of Y satisfy the recursion

Outline of proof: That the expected value is (1 − p)/p can be shown in the following way. Let Y be as above. Then

$\begin{align} \mathrm{E}(Y) & {} =\sum_{k=0}^\infty (1-p)^k p\cdot k \\ & {} =p\sum_{k=0}^\infty(1-p)^k k \\ & {} = p\left(1-p) \\ & {} =-p(1-p)\frac{d}{dp}\frac{1}{p}=\frac{1-p}{p}. \end{align}$

(The interchange of summation and differentiation is justified by the fact that convergent power series converge uniformly on compact subsets of the set of points where they converge.)

Read more about this topic: Geometric Distribution

Famous quotes containing the words moments and and/or moments:

“Athletes have studied how to leap and how to survive the leap some of the time and return to the ground. They don’t always do it well. But they are our philosophers of actual moments and the body and soul in them, and of our manoeuvres in our emergencies and longings.”
—Harold Brodkey (b. 1930)

“Parenting forces us to get to know ourselves better than we ever might have imagined we could—and in many new ways. . . . We’ll discover talents we never dreamed we had and fervently wish for others at moments we feel we desperately need them. As time goes on, we’ll probably discover that we have more to give and can give more than we ever imagined. But we’ll also find that there are limits to our giving, and that may be hard for us to accept.”
—Fred Rogers (20th century)