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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 $\lambda t$

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  • $\begingroup$ Shouldn't your answer to (a) depend on $n$? $\endgroup$ – Brian Tung Jul 16 '19 at 20:37
  • $\begingroup$ Yeah I think it should be gamma distribution with parameters $n, \lambda$ $\endgroup$ – soobster Jul 16 '19 at 21:09
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(a) Suppose that $N=n$. What is the conditional distribution (name and parameter(s)) of $T$?

(a) Exponential distribution with rate $pλ$

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 $\lambda$.

(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 $[0,t]$?

(b) Poisson distribution with parameter $λt$

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.

(c) Suppose that $5$ shocks occur during $[0,t]$. What is the distribution (name and parameter(s)) of the number of replacements during $[0,t]$?

(c) Binomial distribution ~ $(5,p)$

$\color{green}\checkmark$ That is the distribution for the count of failure causing shocks among $5$ shocks which occur independently with identical rate $p$.

(d) Given that $T=t$, what is the conditional distribution (name and parameter(s)) of $N$?

(d) Poisson distribution with parameter $λt$

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 $[0,t)$ ?

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  • $\begingroup$ So (b) should be Poisson distribution with parameter $p \lambda t$ right? $\endgroup$ – soobster Jul 16 '19 at 23:46
  • $\begingroup$ I don't understand your explanation of the last question, which is exactly the same question I asked in this post. Why do you not take account of the the failure-causing shock? $\endgroup$ – soobster Jul 16 '19 at 23:48
  • $\begingroup$ I understood the second question. I answered in my own post. $\endgroup$ – soobster Jul 16 '19 at 23:54
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    $\begingroup$ Yes, the count for failure-causing shocks in interval $[0,t]$ has a Poisson distribution with rate $p\lambda t$. $\endgroup$ – Graham Kemp Jul 17 '19 at 0:22
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    $\begingroup$ And yes, the reason for not taking it into account is because the two processes are independent, and that the condition is that exactly one of the failure-causing has occurred, $\endgroup$ – Graham Kemp Jul 17 '19 at 0:24

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