What is the correct probability distribution for the following situation:

Let's say you want the distribution of hours of tv watched in a
population. If the mean is high and stddev is low (eg: mean 20 hours,
std dev 2 hours), you can say the distribution is essentially normal.

What if, however, the distribution is shifted left, such that there is
a significant population who watch 0 hours of tv?  eg: Mean =3 hrs,
std dev = 2 hours. What is the name of this distribution?  (It is
assymetric, with some a large number at 0, then going to a peak, and
then trailing off to a tail.)

Also, any links (or, best yet, a pointer to an Excel function) would be appreciated.

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Clarification of Question by lookingforstuff-ga on 31 Jan 2006 05:29 PST

Thanks for the feedback so far. I am not an expert in this, so if what
I am saying is incorrect or nonsensical, my apologies.

My intuition is that this is not a Poisson distribution.  First, a
poisson is discrete, and the distribution I am looking for is
continuous. (You can watch 0 hours, 0.0001, 0.0002, etc.) Second, my
intuition is that you would need both a mean and a stddev (or some
measure of spread) to describe the shape, since different populations
could have different variability around the mean.

It also doesn't seem like clustering, since under my assumption the
only "cluster" is at 0, and that is really just a byproduct of the
truncated nature. The rest would be smooth.

Maybe this is nothing more than a truncated normal. I would have
expected the assymetrical nature to change the shape of the
distribution (kind of skewing it left), rather than merely truncating
it, but maybe not.

Is that the answer? 

Also, I would think my example (#hours of tv watched per week for
different populations, where mean can be high for one pop (and hence
normal), and low for another pop (hence capped at 0 and asymetric) for
another) would have made this easy. What am I missing?

I can't tell you the distribution function without knowing the
distribution function. However, I can tell you about a very common
probability distribution function with the properties you are talking
about. The Poisson distribution is the probability of n successful
trials in N attempts, where each trial has a fixed probability of
success. It comes into play, for example, as the number of counts on a
geiger counter in a fixed period of time when the counts are randomly
distributed in time but occuring at a fixed average rate. The
probability distribution function is given by:

P(n,a) = exp(-a)*a^n/n!

where n is the number of counts, and a is the mean number of counts.
Note that the probability of zero counts is:

P(0,a) = exp(-a),

The standard deviation is:

sigma = sqrt(a)

Thus, if the mean is 4, the standard deviation is 2 (close to your
example), and the probability of zero is about 2%. If the mean and
standard deviation are 1, the probability of zero is about 37%.

This distribution converges on a normal distribution (gaussian) as the
mean gets larger and larger.

You can learn more about Poisson's distribution at:

http://www.itl.nist.gov/div898/handbook/eda/section3/eda366j.htm

http://mathworld.wolfram.com/PoissonDistribution.html

You are asking about two different things.  There are of course,
symmetric and asymmetric distributions.  But when you talk about a
large number of incidents at a single number, that is clustering.  A
distribution alone won't cover that.  As an example, insurance company
liability claim amounts tend to follow some distribution, but there
are large numbers of small "nuisance claims" as they call them, and
clustering at round amounts such as $10,000, $50,000, policy limits,
etc.  The nuisance claims and clustering around round amounts need
special treatment.

One model for such cases is called the truncated normal distribution. 
Consider a regular normal distribution.  It actually has infinite
(though vanishingly small) tails in both directions.  But when we
model many measurements such as height in an adult population, the
possibility of a negative result is eliminated a priori.

So it may be useful in the sorts of cases lookingforstuff-ga asks
about, to model the shape of the "central peak" in the distribution by
way of a symmetric normal curve, but then "truncate" the tail on one
or both sides.  Knowing the precise cut-off (in terms of standard
deviations) for such truncations, the area removed from under the
curve is compensated by scaling the height of the remaining part of
the distribution, so that the total area under the curve remains 1
(for the sake of interpretation as a probability density function).

Suggested search term/keywords:  truncated normal distribution


regards, mathtalk-ga

It looks like kottekoe and I were typing our responses at the same
time.  There are two types of distributions, continuous and discrete. 
The normal distribution is an example of a continuous distribution; it
can have any value over a range.  The Poisson distribution tha
kottekoe mentioned is discrete.  In this case it can only have values
that are non-negative integers.  The Poisson distribution is often
used to count the number of incidents.  Characteristic of the Poisson
distribution, and implicit in kottekoe's remarks, is that the mean and
variance are the same.

Although the Poisson distribution is discrete, it is closely related
to a continuous one called the exponential distribution (or slightly
more general, a gamma distribution).

[Exponential distribution -- NIST]
http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm

[Gamma distribution -- NIST]
http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm

Take a look at the pictures for gamma distribution, esp.


regards, mathtalk-ga

I think the answer you are looking for is an Ehrenberg distribution
(named after Andrew Ehrenberg who used this distribution to analyse
sales data).

It is essentially a two-stage mixture process, and portrays the sample
as a mixture of 'watchers' and 'non-watchers'. First we say "What is
the probability that X watches any television at all?". Let's say this
Probability = p

Then GIVEN that he/she is a 'watcher' - what is the CONDITIONAL
distribution of the number of hours watched? Here the Normal
distribution can be used. (Better than the Poisson, because "number of
hours watched" is a continuous variable, not a discrete variable.)

Of course if you two peaks apart from the one at zero (e.g. peaks at
0, and near 2 and near 20), then you need a more compicated mixture -
possibly TWO Normal distributions. Look in Google for "mixture of two
Normal distributions".

e.g. http://www.itl.nist.gov/div898/handbook/eda/section3/histogr5.htm

Or if you Google on 'Images' at

http://images.google.co.uk/images?sourceid=navclient&ie=UTF-8&rls=GGLG,GGLG:2005-21,GGLG:en&q=mixture%20of%20two%20Normal%20distributions%22&sa=N&tab=wi

you will get lots of lovely illustrations.

I hope this helps.

JOHN BIBBY (York, England)