I stumbled across the following random variable, defined as the integral of a linear pure-birth process i.e. a Yule process:

$$ Z_t = \mathbb{E}\bigg[\int_0^t Y_s ds \bigg| Y_t=k , Y_0=1\bigg] $$ where $(Y_t)_t$ is a Yule process of intensity $\lambda$ and $k$ an arbitrary positive integer.

Are there any results regarding the $Z_t$'s distribution?