Proof that the floor of an exponential random variable is a geometric variable

Question

For any real number $x$,$[x]$ represents the smallest integer greater than or equal to $x$. If $X$ is an exponential random variable with mean $1/K$,show that $[X]$ is a geometric random variable with parameter $p = 1 - e^{-K}$.

How can I prove this?

I know what geometric and exponential distributions are!

This looks like a textbook problem. If so, it needs a self-study tag and the rules for self-study need to 2b followed. — Carl, Aug 13 '17 at 18:47
Are you asking for a proof or just hints to help you get there? We are only suppose to apply hints to self study questions. — Michael R. Chernick, Aug 13 '17 at 19:45
The first half of my answer at https://stats.stackexchange.com/a/136956/919 provides a thorough and rigorous proof. — whuber, Aug 13 '17 at 19:59

Taylor · Accepted Answer · 2017-08-14T16:55:29.827

4

Here are a couple hints: \begin{align*} P([X] = x) &= P(x \le X < x+1) \tag{logic} \\ &= P(x < X \le x+1) \tag{$X$ is a continuous rv} \end{align*} for any non-negative integer $x$.

edited Aug 14 '17 at 16:55

answered Aug 14 '17 at 00:13

Taylor

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Proof that the floor of an exponential random variable is a geometric variable

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