Let $N(t)$ be $PP(λ)$.Given that $N(t)=n$, compute the probability of
a) Last event before $t$ occurs before $3t/4$.
b) First event after $t$ occurs after $t+h$, $0<h$.
c) $S_1<2$, $S_3$>4 for $t=10, n=3$.
I know I should use order statistics to compute these probabilities. But cannot figure out how.
a) $P(S_{N(t)}<{3t}/4|N(t)=n)$
$={P(S_{N(t)}<{3t}/4, N(t)=n)}/P(N(t)=n)$
$=P(S_{N(t)}<{3t}/4)$
b)$P(S_{1}>h|N(t)=n)$
c)$P(S_{1}<2, S_3>4|N(10)=3)$
Also, any good resource with many solved examples on Poisson processes and Continuous time Markov Chains will be highly appreciated.