I have a random variable which can have any value from the set of natural numbers. For example, the probability of the random variable having a low value is higher than the probability of the random variable having a higher value.

So would this be a continuous probability distribution or a discrete probability distribution?

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    $\begingroup$ Do you have any ideas as to which it is and why? $\endgroup$ – Henry Sep 29 '15 at 18:17
  • $\begingroup$ I believed it would be a continous distribution because the probability would be a continuous value? does that make sense? $\endgroup$ – andrew Patterson Sep 29 '15 at 18:19
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    $\begingroup$ What is your definition of a continuous probability distribution? What difficulty are you having in applying it to this situation? BTW, much of your question makes no sense at all--please re-read the passage following "for example." $\endgroup$ – whuber Sep 29 '15 at 18:27
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    $\begingroup$ @Henry That does not seem like a distinguishing factor, because every continuous variable does have a probability assigned to every possible value it can take. Perhaps you meant to stipulate "nonzero probability"? But in that case many discrete variables would not actually qualify as discrete, either. $\endgroup$ – whuber Sep 29 '15 at 18:36
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    $\begingroup$ @whuber - I see your point. Perhaps I should have said that for a discrete distribution these probabilities add up to $1$ while for a continuous distribution they are all $0$ (for intermediate cases the distribution may be a mixture) $\endgroup$ – Henry Sep 29 '15 at 19:12

By definition your distribution is discrete, because you can obtain all the values by counting.

Your confusion may stem from two sources. One is that often people assume that discrete also means finite. This is not true, e.g. the Poisson distribution is defined on the non-negative integers, which is an infinite countable set $[0,+\infty)$.

Another source could be usage of computer-generated random numbers, or pseudo random number generators. Since in computers even continuous variables, such as real numbers, are represented by countable sets (e.g. IEEE double precision floating point), the PRNGs generate discrete sequences to approximate continuous variables. So, in some sense, everything we do in computers is discrete.

By "computers" I mean digital computers. Analog computers are different.


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