Size-biased probability generating function
= Size-biased probability generating function
{title2=$G_{\widetilde Y}(s)=sG_Y'(s)/\mathbb EY$}
For a positive <integer>-valued <random variable> with finite mean $\mu$, the <size-biased distribution> has probabilities $k\mathbb P(Y=k)/\mu$. They sum to one by the definition of <expected value>. Multiplying the differentiated <probability generating function> by $s/\mu$ gives the displayed identity for $0\leq s<1$, with the endpoint interpreted by its limit.