Let $N(t)$ be Poisson process with intensity $\lambda$, and let $Y_{1}, Y_{2}, \ldots$, be independent and identically distributed random variables with cumulative distribution function $$ G(z)=z^{\alpha} \quad \text { for } \quad z \in[0,1]. $$ Let $2(t)=\min \left\{y_{1}, Y_{2}, \ldots, Y_{N(t)} ight\}$ (a) Determine $\mathbf{P}(Z(t)>z \mid N(t) \geq 1)$. (b) Describe the behavior of this probability for large $t$. Hint: consider separately $z>0$ and $z=0$. CS.VS. 1263|