Size-biased probability generating function ($G_{\tilde{Y}}(s)=s{G}_Y^{\prime }(s)/\mathbb{E}Y$)

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 $0\le s<1$, with the endpoint interpreted by its limit.