I think $Y$ is of a discrete distribution since it takes only an integer and I tried to use the cumulative distribution $X$ to find the distribution of $Y$ but it seems not possible.
How should I approach this?
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What have you tried so far? Can you express the set ${Y = k}$ in terms of $X$? Perhaps you can take it from there. – Mushu Nrek Jan 09 '24 at 19:38
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Does this answer your question? https://math.stackexchange.com/q/805796/874549 – noam.szyfer Jan 09 '24 at 19:40
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Can you tell what approach did you try? – Ajin Shaji Jose Jan 09 '24 at 19:41
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@MushuNrek thanks for that! My mistake was trying to integrate to infinity – Ali Sabi Jan 09 '24 at 19:56
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@noam.szyfer yes! thanks – Ali Sabi Jan 09 '24 at 19:56
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Let $X \sim Exp(\lambda),$ so that $X$ is distributed according to the pdf $$f_\lambda(x) = \lambda e^{- \lambda x} \chi_{[0, \infty)}(x)$$ and $Y := \lfloor X \rfloor.$ Then $$\mathbb{P}(Y = k) = \mathbb{P}(k \leq X < k + 1) = \int_k^{k + 1} f_\lambda(x) dx = e^{- \lambda k} - e^{- \lambda(k + 1)} = e^{- \lambda k}(1 - e^{- \lambda}).$$ I hope this helps. :)
Actually Fritz
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