# How to figure out this problem

My apologies if this comes out as stupid.

Problem (Domain : stock exchanges): I have been given the sum of orders placed per instrument(security or company) for about 25000+ instruments. So my dataset looks something like this:

[Instrument] [Order count]   [Average Order Size(AOS)]  [Orders above 10*AOS]

AAA            20                 10000.0                 ?
BBB            5000               24334.5                 ?
...................


I know the average order size placed for an instrument as shown above. I want to calculate an estimate for the number of orders placed which had a order size above 10*[average order size] per instrument. I don't have an idea about the distribution. Assuming it's a normal distribution, I don't have an idea about the standard deviation. Is there any way to get a practical answer (I don't care if it is a value in a range.) for my problem? I hope this question is relevant here. Thanks.

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:(. Someone help! – nakiya Feb 14 '11 at 20:17
There's simply not enough data out there to make an inference. Do you have any past data that you can harvest to get more insight on the distribution? – Yuval Filmus Feb 14 '11 at 21:40
What about Markov's inequality? You know that order sizes are always positive, so that should do it. – Raskolnikov Feb 14 '11 at 21:55

Given that the order size is positive, You could use Chebyshev's inequality or Markov inequality as people call it. This does not have any assumption on the underlying distribution. If you know the underlying distribution, then you could come up with a tighter bound.

In your case, $P(\text{Order size} > 10 \times \text{AOS}) \leq \frac{1}{10}$

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Thanks. That inequality helps a lot. I wish I knew what the distribution is and can get a tighter bound. But thanks anyway. – nakiya Feb 15 '11 at 10:07

It is quite unlikely to be a normal distribution. There is the heuristic 80/20 rule-80% of your orders come from 20% of the customers.

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Assuming a normal distribution, and given that you only known the average value, there is no way of estimating the standard deviation and thus the number of outlier values (values with size greater than ten times the average).

This is not the case, however, for other families of distributions. There are, for instance, families of distribution depending on only one parameter which therefore determines both the expectation value and the variance.

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Apparently I've misread the quesiton. – Rasmus Feb 16 '11 at 22:25

OK, I already suggested using Markov's inequality, but this only gives a very rough estimate for an upper bound on the probability of having an order of large size. Also see Sivaram's explanation.

But maybe what you really ought to do is model the distribution of the order sizes. I think it's not unreasonable to assume some exponential distribution which depends only on one parameter, the average order size. Or maybe a power law distribution, Pareto type, it would give rise to 80-20 type rules. If you possess historical data, you can try to check how well your data fit this model.

Maybe you should provide us the context of the question. Is it homework? If so, for what discipline?

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