A system receives shocks according to a Poisson process with rate $λ$. Each shock independently causes the system to fail with probability $p$. Let $T$ denote the failure time of the system, and let $N$ be the number of shocks received up to and including time $T$.
(a) Suppose that $N = n$. What is the conditional distribution (name and parameter(s)) of $T$?
(b) Suppose that a failed system is always immediately replaced by a new system. What is the distribution (name and parameter(s)) of the number of replacements that occur during a fixed time interval $[0, t]$?
(c) Suppose that $5$ shocks occur during $[0, t]$. What is the distribution (name and parameter(s)) of the number of replacements during $[0, t]$?
(d) Given that $T = t$, what is the conditional distribution (name and parameter(s)) of $N$?
This is a question from my university's past final exam. I am not too sure if my answers are correct, please verify:
(a) Exponential distribution with rate $p\lambda$
(b) Poisson distribution with parameter $\lambda t$
(c) Binomial distribution ~ $(5,p)$
(d) Poisson distribution with parameter $\lambda t$
Best Answer
Under the condition that $N=n$, no sock until the $n$-th cause the system to fail, and so $T$ measures the time until the $n$-th shock; ie, its distribution is that for the sum of $n$ exponential random variables each with rate $\lambda$.
That is the distribution for the count of shocks within the interval.
We seek the distribution for the count of failure causing shocks within the interval.
$\color{green}\checkmark$ That is the distribution for the count of failure causing shocks among $5$ shocks which occur independently with identical rate $p$.
At time $T$ the first failure-causing shock has occurred, and $N-1$ counts the shocks which have not caused failure.
What is the distribution for the count of non-failure-causing shocks within period $[0,t)$ ?