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A random process $X(t)$ is defined as $X(t)=1, 0\leq t \leq Y$ and $0$ otherwise where $Y$ follows an exponential distribution. What is the $pmf$ of $X(t)$

I am a bit confused on the definition of $pmf$ for random process. By definition, the pmf of $X(t)$ is $$Pr(\{ X(t, \xi)=1 \})=Pr(\{ (t, \xi) | 0\leq t \leq \xi \})=\int_0^{\infty}\int_0^\xi f(t,\xi)dtd\xi$$

However, I am confused how I can calculate $f(t, \xi)$. I know from the question that $f(\xi) = \lambda e^{-\lambda \xi}$ where $\lambda$ is the parameter of the geometric distribution. But I don't see how to get $f(t,\xi)$

AspiringMat
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  • My guess is that the question asks you to find the distribution of $X(t)$ for a fixed $t$ (and this is trivial). The distribution of the entire process is a measure on an infinite dimensional function space. – Kavi Rama Murthy Nov 27 '18 at 09:13
  • @KaviRamaMurthy That makes a lot more sense now. Just integrate from $t$ to infinity the geometric pdf. This is one of the reason I don't like the notation of suppressing the sample space in random processes. It just makes things more confusing for learners such as myself. Thank you! – AspiringMat Nov 27 '18 at 09:22

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