I'm reading a definition in DeGroot's book that begins with the statement:

"Let X be a bounded discrete random variable whose p.f. is f."

Then he goes on to define the expectation of X.

However, I cannot find a definition of what is meant by a "bounded" discrete random variable anywhere in the book (after an hour of looking). I do know what a discrete random variable is, but what does the word "bounded" mean in this case?


  • 1
    $\begingroup$ Bounded just means that there exist a number $M>0$ such that $|X| \le M$ with probability 1. $\endgroup$ Jun 2, 2014 at 19:10

1 Answer 1


Bounded, in this case, means what it means in pretty much every mathematical context - there's a maximum and minimum that the variable never exceeds. If you think about the normal random variable, Z, technically Z could be anything, it's just that as you get farther and farther away from zero, the probability diminishes exponentially.

So, $P(Z > z)$ is never actually equal to zero for any finite z. [EDIT] This means that Z is unbounded.

In a bounded random variable, X, there's some value m and some value M such that:

$P(x < m) = 0$ and $P(x>M) = 0$

  • $\begingroup$ And this means that the normal random variable Z is unbounded? $\endgroup$
    – David
    Jun 3, 2014 at 16:08
  • $\begingroup$ That's correct. Z is unbounded. Edited my answer to make that clear. Z is also different because it's not discrete, but you already mentioned that you know what a discrete RV is. $\endgroup$
    – Duncan
    Jun 3, 2014 at 16:33

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