This is with ref to stock market figures. 
Lets say the index
(whichever) has moved up at an average rate (compound) of 10% per
annum. If we have the figures available for the various stocks (daily)
as well as for the index as a whole, how would one check whether the
stocks rise was or was not 'random' (to the extent of tech analysis
not being possible)

One could, for example, take the initial share price figures of ten
years ago and generate the extraploated, periodic (say daily or
monthly)  prices on a basis of 10% p.a.. Then one could test these
figures against the actual share prices at the same periods.  But what
sort of tests does one do to find out if the figures meet a randomness
criteria (the philosophical question of whether the figures are
'truly' random may be put aside for now, just whether they seem
reasonably random according to current methods)
Typically, there are long periods where share prices increase very
frequently and other long
periods where they decrease and this seeming pattern would have to be accounted for

Should one try to fit the data to a poisson distribution, normal dist
or binomial dist, and how should it be done e.g should one calculate
the percentage changes from day to day and expect these to be
distributed normally about the average actual daily percent change?
How would one use a Poisson dist or a bin one? Would it be even make
sense to do this?

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Clarification of Question by myq-ga on 02 May 2005 10:29 PDT

e.g if one were given the numbers 1, 2, 5, 4, 8, 10, 3, 6, 2, 5, 7, 1
, 3, 8, 4 how would one go about finding out what the chance was that
they were randomly generated from 1 to 10.
A series of all 15  1's would be as likley to obtain from random
generation as the above specific series BUT GIVEN the 1's the chance
of it being random would be a LOT LESS than the chance of the above
series.
How does one calculate the 'LOT LESS'?

Asking whether something is random or not doesnt make sense...

Statistical tests are testing whether a model fits or not... you cant
test 'randomness'.

To be honest what you want in general isnt clear either.  Maybe you
could clarify what you want.

Do you mean that you want to test whether a increase/decrease is
attributed to a shift in the mean or if its just random?

I believe there are statistical tests to show that certain data is
CONSISTENT with randomness (that does not mean that it really is
random, of course). For example, though it is possible to get 100
Heads in 100 coin tosses, if you were told someone had got 100 heads
in a row you would expect that there was a problem with the coin, or
something else.
In this case, one could try the model that there was a 55% chance that
the next 1 cent change in a share price would be upwards and 45% that
it would be downwards. One could extrapolate the prices for each day
starting with the known actual prices THEN compare with the actual
daily prices. Then, I am supposing, there would be relatively
straightforward statistical tests to determine within certain
tolerance levels whetrher the actual prices  being random was
consistent with the data.
Statistics is not my field but I do know definitivel that various
researchers have studied this problem and most have come back saying
the data is consistent with the theory that prices move randomly.
Look forward to someone explaining in greater detail how one could
show such consistency (which agai, is not proving the theory)

What you're looking for is a way to quantify the information content
(i.e. deviation away from randomness) in a set of what may look like
"noisy" data.

Someone by the name of Shannon introduced such a method many years ago.
Shannon's information theory was formulated in 1948 (!), but it still applies
to this day and the theory has been put to practical use in order to
measure the information content in neuronal spikes (brain cells send
information that looks very noisy (like stock data), but is actually
rich in content).

(I had a pdf of Shannon's seminal paper somewhere but I can't seem to find it)

Nevertheless, what you're looking for is possible and while I was
about to write a page-long reply to your query, I believe the link
below may provide
you with more than enough information (practical, as well as theory).

You can also look at the deviation from the geometric mean as a way to
quantify 'noise'...

http://survivor99.com/lcg/english/portfolio/ENG_page.html
It would be quite straightforward to implement most of these
techniques by using a Matlab script.

Linear systems theory is very applicable to your query...

hope this helps you!