2

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!

Silverfish
  • 20,678
  • 23
  • 92
  • 180
Vishal
  • 23
  • 1
  • 4
  • 3
    This looks like a textbook problem. If so, it needs a `self-study` tag and the rules for [self-study](https://stats.stackexchange.com/tags/self-study/info) need to 2b followed. – Carl Aug 13 '17 at 18:47
  • 1
    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
  • 2
    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

1 Answers1

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 $x$.

Taylor
  • 18,278
  • 2
  • 31
  • 66