(Original post by TAEuler)
Okay, so what you're saying is:
The ability to reexpress continuous random variables is dependent on the step size of the discrete random variable. In my example, the step size is 1. This is a very small step size which means that the discrete curve becomes very close to the continuous curve?
But i'm not really sure how a small standard deviation relates to that? And what is classed as a small one?
The point is that 1 is only small relative to the spread of the distribution (we have values going from 0 - 100). If the spread was only (0-1), obviously a step size of 1 would be pretty rubbish.
It's easier (but technically "wrong") to look at things the other way around - we've been talking about approximating a discrete distribution with a continuous one, but instead imagine approximating a continuous one with a discrete one.
To talk about something slightly more concrete; imagine approximating a semicircle with the squares on graph paper. If your semicircle is large relative to the squares on the graph paper, it's a good approximation - if the diameter is only a few times greater than the width of a square, the approximation is going to be poor.
The size of the squares corresponds to the 'step size'; the diameter corresponds to the standard deviation (recall the standard deviation is a measure of the "spread" of the distribution).
Note that "small relative to the standard deviation" is a bit of a judgement call. 20% of the standard deviation is obviously "not small", and 0.01% almost certainly does count as "small". But exactly what is "small enough" is (IMHO) a bit of a judgement call.
It's also worth noting that "relative to the standard deviation" is more a rule-of-thumb than a completely reliable measure. If you imagine a game where I toss a coin, and give you £90 if it's heads, £10 if it's tails, and then throw a dice 5 times and give you the total thrown in pennies, then the distribution will have 2 very sharp peaks 25 pence wide centered at £90.15 and £10.15. The standard deviation is going to be £20, but a step size of 20p isn't going to give you any kind of accurate feel for the distribution.