A rodent has a lifetime of T days where T is exponentially distributed with parameter λ . It follows a Poisson process. Its predators pass at a rate of δ per day and each predator can catch with probability p. What is the point probability the rodent dies a natural death?
How can I go about this question? Is this a joint distribution?
One way to think about it is to say that, if we pretend the rodent has an infinite lifetime for a moment, then there is a geometric random variable say $X$ that determines the number of attempts that the predators need to make to kill the rodent. In reality the Bernoulli process associated to the predators hunting the rodent may be cut short by the rodent dying of natural causes, but thinking this way allows us to separate the "natural causes" process from the "predation" process.
Following this reasoning, we conclude that the rodent will be killed by a predator at time $\delta X$ if $T>\delta X$ and it will die of natural causes at time $T$ if $T<\delta X$. Thankfully $T=\delta X$ has probability zero so we don't have to ponder how to properly interpret that case.
Now use the total probability formula and the assumption that $T$ and $X$ are independent to marginalize over the distribution of $X$, which will allow you to complete the problem.