I need to calculate $P[N_{s}=k||N_{u},\,u \geq t]\,(s \leq t)$ for the Poisson process. However, I have been instructed to use the following example in order to do so: Example: The Poisson process $[N_{t}:t \geq 0]$ has independent increments. Suppose that $0 \leq t_{1} \leq \cdots \leq t_{k}...

Let $R$ be a commutative ring with identity, $I$ an ideal in $R$, $\pi : R \to R/I$ the canonical projection, and let $S$ be some multiplicative subset of $R$. Define $\theta : S^{-1}R \to (\pi(S))^{-1} (R/I)$ by $\theta (r/s) = \pi(r)/\pi(s)$. Prove that $\theta$ is an epimorphism with kernel...

For a random variable $T \geq 0$ with distribution function $F$, a real number $t > 0$ and $\mathcal{G}= \{ \emptyset, \{ T > t \}, \, \{ T \leq t \}, \, \Omega \}$, I need to evaluate the conditional expectation of $E|T-t||\mathcal{G}$, where $\mathcal{G}$ is my sub-$\sigma$-algebra. Now, I as...

Let $R$ be a commutative ring with identity, $I$ an ideal in $R$, $\pi : R \to R/I$ the canonical projection, and let $S$ be some multiplicative subset of $R$. Define $\theta : S^{-1}R \to (\pi(S))^{-1} (R/I)$ by $\theta (r/s) = \pi(r)/\pi(s)$. Prove that $\theta$ is an epimorphism with kernel...