To find the moment generating function (MGF), mean, and variance of a random variable $X$ that follows a geometric distribution with probability mass function $P(X = x) = q^{x-1} p$, where $p$ is the probability of success and $q = 1 - p$ is the probability of failure, we can proceed as follows.

1. Moment Generating Function (MGF)

The MGF of a random variable $X$, $M_X(t)$, is defined as: $M_X(t) = E[e^{tX}]$ For the geometric distribution, we calculate $M_X(t)$ as follows:

$M_X(t) = \sum_{x=1}^{\infty} e^{tx} P(X = x)$ Substitute $P(X = x) = q^{x-1} p$: $M_X(t) = \sum_{x=1}^{\infty} e^{tx} q^{x-1} p$ Factor out $p$: $M_X(t) = p \sum_{x=1}^{\infty} (e^t q)^{x-1} e^t$ Let $y = e^t q$. Then the series becomes: [ M_X(t) = p e^t \sum_{x=