From: horst.kraemer@snafu.de (Horst Kraemer)
Newsgroups: alt.sci.math.probability,alt.sci.math.statistics.prediction,sci.math.num-analysis,sci.stat.math
Subject: Re: Scaling of probability curve?
Date: Tue, 22 Dec 1998 19:22:56 GMT
On 22 Dec 1998 14:31:31 GMT, "Abbot Cooper"
wrote:
> I have a known historical distribution of patients, expressed as a
> probability "curve" of having X number of patients arrive on any given day
> to a certain area of the hospital. For instance, the following:
>
> 0 patients = 25%
> 1 patient = 50%
> 2 patients = 25%
> If you add up the probabilities for each of these (1 * .50) + (2 * .25),
> you get an overall average of 1 patient per day. The question that I am
> trying to answer is, how can I come up with a new "curve" for a predicted
> change in the overall volume. I have done this for some simple examples
> such as above, but don't know how to obtain a formula to work in any
> instance or for more complex curves. For instance, one of the example I
> _can_ do is to double the above curve to the following:
A "reasonable probability distribution" for a given average m (of
patients per day) of events which are "infrequent" in a certain sense
is the POISSON distribution.
m^k
Pr { n = k } = ---- * exp (-m)
k!
For m = 1 the probabilities will be
0 0.36788
1 0.36788
2 0.18394
3 0.06131
4 0.01533
5 0.00307
...
m=1.5
0 0.22313
1 0.33470
2 0.25102
3 0.12551
4 0.04707
5 0.01412
...
m=2.0
0 0.13534
1 0.27067
2 0.27067
3 0.18045
4 0.09022
5 0.03609
...
They add up to 1 if you take the limit n->oo.
This model fits many phenomena surprisingly well, like the number of
telephone calls within a certain period of time, the number of persons
arriving at a counter asking for service etc...
A famous application of this distribution was a comparision of the
number of persons killed by a kick of a horse in a polish cavalry
regiment within subsequent years in the beginning of the century.
Regards
Horst