D1 - higher moments

moment

for a RV $X$ , the $k^{t h}$ moment of $X$ is $E [X^{k}]$

so, $E [X]$ is the first moment of $X$
let $X \sim$ geometric $(p)$

E [X^{2}] = \sum_{i = 1}^{\infty} i^{2} p_{X} (i) = \sum_{i = 1}^{\infty} i^{2} (1 - p)^{i - 1} p

since the second moment is an expectation, it can be computed using conditioning:

\begin{aligned} E [X^{2}] & = E [X^{2} | first flip is head] \cdot p + E [X^{2} | first flip is tail] \cdot (1 - p) \\ = 1^{2} \cdot p + E [(1 + X^{2})] \cdot (1 - p) \\ = p + E [1 + 2 X + X^{2}] \cdot (1 - p) \\ = p + (1 + 2 E [X] + E [X^{2}]) (1 - p) \\ = p + (1 - p) + 2 (1 - p) \frac{1}{p} + E [X^{2}] (1 - p) \\ p E [X^{2}] & = 1 + 2 (\frac{1 - p}{p}) \\ E [X^{2}] & = \frac{2 - p}{p^{2}} \end{aligned}

here, $E [X^{2} | first flip is tail] = E [(1 + X)^{2}]$
similarly, $E [X^{3}] = 1^{3} \cdot p + E [(1 + X)^{3}] (1 - p)$