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Due to the stress of coping with business, Harry begins to experience migraine headaches of random severities. The times when headaches occur follow a Poisson processes of rate $\lambda$. Headache severities are independent of times of occurrence and are independent, identically distributed random variables $\{H_{i}\}$ with common exponential distribution $$P\left[H_{i}\leq x \right] = 1 - \exp\{-x\}.$$ (Assume headaches are instantaneously and have duration zero.)

Harry decides he will commit himself to the hospital if a headache of severity greater than $c>0$ occurs in the time period $\left[0,t \right]$. Compute

$$ P\{ \text{Harry commits himself in } \left[ 0, t \right]\}.$$

Here I was trying to compute the probability if the complementary event that Harry won't commit himself in $\left[0,t \right]$ and I think that probability of not committing is $P\left[H_{i}=0 \right]$ recalling that $\max\{a,b\} \leq c$ iff both $a \leq c$ and $b \leq c$.

Alexis Sandoval
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1 Answers1

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The probability that Harry will experience precisely $k\in\mathbb Z_+$ migraine attacks in $[0,t]$ is $(\lambda t)^k\exp(-\lambda t)/k!$ Conditional on $k$ realizations, the probability that none of these $k$ attacks is severe enough for Harry to get admitted is $[1-\exp(-c)]^k$. Therefore, using the law of total probability, one can compute that the probability of Harry not going to the hospital in $[0,t]$ is $$\sum_{k=0}^{\infty}\frac{(\lambda t)^k}{k!}\exp(-\lambda t)[1-\exp(-c)]^k=\exp[-\exp(-c)\lambda t].$$

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