I have to prove that the function $$ \rho(s)=\frac{2}{(2-s)(3-s)}, \ 0\leq s\leq 1$$ is a probability generating function of a distribution. Calculate this distribution and its mean, variance, and standard deviation.
I try first use partial fraction, obtaing that $$ \rho(s)=\frac{2}{(2-s)(3-s)} =\frac{2}{(s-3)(s-2)} =\frac{A}{(2-s)}+\frac{B}{(3-s)} $$ with some calculations I obtained that $A=2$ and $B=-2$ then $$ \rho(s)=\frac{2}{(2-s)(3-s)} = \frac{2}{(s-3)}-\frac{2}{(s-2)} $$
but How can I applied the definition of Probability generating function of a distribution ?
Someone can help me to solve this pls. Thanks for your time and help everyone.
Hint: Use the geometric series:
$$\frac{1}{s-2} = -\frac{1}{2} \cdot \frac{1}{1-\frac{s}{2}} = - \frac{1}{2} \sum_{k=0}^{\infty} \left( \frac{s}{2} \right)^k.$$
Derive a similar expression for $\frac{1}{s-3}$. Plugging this into your computations allows you to determine the probability mass function (and, thus, the distribution).