# Finding Conditional Distribution from Poisson Distribution

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 $$\lambda t$$

• Shouldn't your answer to (a) depend on $n$? – Brian Tung Jul 16 '19 at 20:37
• Yeah I think it should be gamma distribution with parameters $n, \lambda$ – soobster Jul 16 '19 at 21:09

(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 $$[0,t)$$ ?

• So (b) should be Poisson distribution with parameter $p \lambda t$ right? – soobster Jul 16 '19 at 23:46
• 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? – soobster Jul 16 '19 at 23:48
• I understood the second question. I answered in my own post. – soobster Jul 16 '19 at 23:54
• Yes, the count for failure-causing shocks in interval $[0,t]$ has a Poisson distribution with rate $p\lambda t$. – Graham Kemp Jul 17 '19 at 0:22
• 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, – Graham Kemp Jul 17 '19 at 0:24