I have never taken probability theory, and I wonder whether one can express some random variable $X$ with mean $\mu$ and variance $\sigma ^2$ in terms of $\mu$ and $\sigma$ only. Or at least something close to it.
Thank you for your help.
No - the mean and variance (if they exist) tell you about the location and scale of a distribution but nothing about its shape.
If a random variable $X$ has mean $\mu$ and variance $\sigma^2$ then the random variable $Y$ defined by $Y=aX+b$ for real $a$ and $b$ has mean $a\mu+b$ and variance $a^2 \sigma^2$.
So given a particular shape of distribution (well behaved enough to have a mean and variance), it is possible to find a distribution which has the same shape but any mean and (positive) variance you specify.
A Poisson random variable has mean lambda and variance lambda. In the case of one parameter distirbutions the mean and variance will both be functions of a single parameter.