Proving a Geometric Progression Formula, Related to Geometric Distribution

Question

I am trying to prove a geometric progression formula that is related to the formula for the second moment of the geometric distribution. Specifically, I am wondering where I am going wrong, so I can perhaps learn a new technique.

It is known, and I wish to show: $$ m^{(2)} = \sum_{k=0}^\infty k^2p(1-p)^{k-1} = p\left(\frac{2}{p^3} - \frac{1}{p^2}\right) = \frac{2-p}{p^2} $$

Now, dividing by $p$ both sides, and assigning $a = 1-p$ yields: $$ \sum_{k=1}^\infty k^2a^{k-1}=\frac{2}{(1-a)^3}-\frac{1}{(1-a)^2} \qquad \ldots \text{(Eq. 1)} $$

I want to derive the above formula. I know: $$ \sum_{k=0}^\infty ka^{k-1}=\frac{1}{(1-a)^2} $$ Multiplying the left side by $1=\frac aa$, and multiplying both sides by $a$, $$ \sum_{k=0}^\infty ka^k = \frac{a}{(1-a)^2} $$ Taking the derivative of both sides with respect to $a$, the result is: $$ \sum_{k=0}^{\infty}\left[a^k + k^2 a^{k-1}\right] = \frac{(1-a)^2 - 2(1-a)(-1)a}{(1-a^4)} = \frac{1}{(1-a)^2}+\frac{2a}{(1-a)^3} $$ Moving the known formula $\sum_{k=0}^\infty a^k = \frac{1}{1-a}$ to the right-hand side, the result is: $$ \sum_{k=0}^\infty k^2 a^{k-1} = \frac{1}{(1-a)^2} + \frac{2a}{(1-a^3)} - \frac{1}{1-a} $$

Then, this does not appear to be the same as the original formula (Eq. 1). Where did I go wrong?

Thanks for assistance.

Answer 1

You've actually almost arrived at the correct solution - your mistake is in the differentiation with respect to $a$ - the LHS is $\sum_{k=0}^{\infty}\left[a^k + k^2 a^{k-1}\right]$ instead of $\sum_{k=0}^{\infty}\left[k^2 a^{k-1}\right]$.

You're finished because $\frac{1}{(1-a)^2} + \frac{2a}{(1-a)^3} = \frac{a+1}{(1-a)^3} = \frac{2}{(1-a)^3} - \frac{1}{(1-a)^2}$.

EDIT: Here's some generalization: In general, $\sum_{n \ge 0} \binom{n}{k}x^n = \frac{x^k}{(1-x)^{k+1}}$, which is basically proved by differentiating $\sum_{n\ge 0} x^n=\frac{1}{1-x}$ $k$ times, multiplying by $x^k$ and dividing by $k!$.

From this one can calculate $\sum_{n \ge 0}n^k x^n$ by expressing $n^k$ as a combination of binomial coefficients: $n^k = \sum_{i=0}^{k} \binom{n}{i} a_{i,k}$. If you interpret $n^k$ as the number of functions from a $k$-set to an $n$-set, you realize that $a_{i,k}$ count the number of surjective function from a $k$-set to an $i$-set.

This allows you to compute the $k$'th moment of the geometric distribution quite easily, and it will always be of the form $\frac{f(p)}{p^k}$ where $f$ is a polynomial of degree $k-1$.

EDIT 2: It can be seen that $f(0)=a_{k,k}=k!, f(1)=a_{1,k} = 1$. This allows you to write $f(x)=x+k!(1-x)+(x-1)xg(x), \deg g < k-2$, which gives you the second moment immediately ($k=2$): $f(p)=p+2!(1-p)=2-p$.

In general, $f(p)=\sum_{i=1}^{k} a_{i,k} (1-p)^{i-1}p^{k-i}$. Note that you can express $a_{i,k}$ with Stirling numbers of the second kind: $a_{i,k}=i! \lbrace\textstyle{k\atop i}\rbrace= \sum_{j=0}^{i} (-1)^{i-j}\binom{i}{j}j^k$.

Compare with my edit to this answer.

Answer 2

You have $$\sum_{k=0}^{\infty} ka^k = \dfrac{a}{(1-a)^2}$$ Differentiating with respect to $a$ gives us $$\sum_{k=0}^{\infty} k^2 a^{k-1} = \dfrac{(1-a)^2 - a \times 2 \times (a-1)}{(1-a)^4} = \dfrac{1-a + 2a}{(1-a)^3} = \dfrac{a-1+2}{(1-a)^3}\\ = \dfrac2{(1-a)^3} - \dfrac1{(1-a)^2}$$