Normal Distribution

[Chintan]

Hi David,

I was watching your 2nd movie on Quant and I have a question relating to normal distribution therein.

You mentioned that many dice together can be construed as a normal distribution.

Till now, we've always interpreted normal distribution as the one which is "continuous" and not discrete. Continuous will imply infinite set of data which needs to be "measured" since it cannot be counted.

However, when we mention many dice, will that not be finite set of data and outcomes which can be counted. So, why do we call it as a normal distribution.

Please help clear my doubt / confusion.

Thanks in advance.
Chintan

 

[David Harper CFA FRM]

Hi Chintan,

It is a great observation. I hope I was careful in my language. Because strictly speaking, you are correct: the distribution of dice will always be discrete - these outcomes can always be counted.

The reference to normal here regards the central limit theorem, which says the sample average (as itself a random variable) of a large roll of dice tends to be normally distributed (and the summation, too, by extension). The greater the number of dice we roll, the more normal the sampling distribution (of sample means) becomes.

To your point: this distribution only "tends toward" or "converges on" the continuous normal. It is a discrete approximation of continuous normal. This is not uncommon: both the binomial and the Poisson (which are discrete distributions) tend toward the continuous normal without becoming continuos normal (binomial, if large N; Poisson, if large lambda). In all cases here, it is about a distribution of dots that looks pretty much like a continuous line.

The hardest thing about the CLT, IMO, is getting the mind around: it is about the distribution of sample means. In the case, a single die is UNIFORM distrete; the average of several die is a sample. Take another. And another. The CLT refers to the distribution of these several samples. Hope that helps...

David