Suppose you are watching a radioactive source that emits particles at a rate described by the exponential density with $\lambda=1$
The probability $P(0,T)$ that a particle will appear in the next T seconds is $P ([0, T ])$ = $\int_0^T\lambda e ^{-\lambda t}$
Find the probability that a particle (not necessarily the first) will appear after 4 seconds from now.
How can I setup my equation? My hunch is that this has something to do with the memoryless property of the exponential r.v.
Would appreciate any guidance. thanks
NOTE: The answer given is 1
If you wanted to find the probability that the first particle would appear after $4$ seconds from now, I think it would be this: $$1-\int_0^4 \lambda e^{-\lambda t} \,\mathrm{d}t$$
But you are asked only the probability that some particle will appear after $4$ seconds from now is just a certainty, or $1$. This is because whenever a particle spawns the probability resets (is this the memorylessness you were talking about?).