# Distribution of a linear pure-birth process' integral

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?

• Are you sure you want to condition on $Y_t=k$ rather than $Y_0=k$? – Yuval Peres Jun 21 at 7:31
• Yes. Although I should have mentioned that the Yule process starts in $Y_0=1$. I updated the post. – Gabriel Jun 21 at 9:27
• Where is the random variable here? Are you interested in the conditional expectation of the integral, or in its conditional distribution? – Kostya_I Jun 21 at 10:06
• The random variables are the Yule process $(Y_t)_t$ and the conditional expectation $(Z_t)_t$. I am interested in the distribution of the conditional expectation above. I think intuitively one might compare it to the integral of a Brownian bridge (although BB is not non-negative and increasing). – Gabriel Jun 21 at 11:35
• The conditional expectation is a deterministic function of $k$. – Yuval Peres Jun 21 at 12:40