- Distributions and characteristics thereof

- Importance of error distribution analysis
- Confidence levels

(Entire section in one PDF file).

For a given rate of emitted photons, there's a probability function which
gives the number of photons we detect, even assuming 100% detection
efficiency, because of *statistical* uncertainties. In addition,
there may also be *instrumental* uncertainties. Consequently, we
now turn to the concepts of probability distributions, with particular
interest in the distribution which applies to the detection of photons.

- concept of a distribution : define
*p*(*x*)*dx*as probability of event occuring in*x*+*dx*:*p*(*x*)*dx*= 1

Some definitions relating to values which characterize a distribution:

mean≡*μ* = *xp*(*x*)*dx*

variance≡*σ*^{2} = (*x* - *μ*)^{2}*p*(*x*)*dx*

standarddeviation≡*σ* =

median : mid-point value.

=

mode : most probable value

Note that the geometric interpretation of above quantities depends on the nature of the distribution; although we all carry around the picture of the mean and the variance for a Gaussian distribution, these pictures are not applicable to other distributions, but the quantities are still well-defined.

Also, note that there is a difference between the *sample* mean,
variance, etc. and the *population* quantities. The latter apply
to the true distribution, while the former are estimates of the latter
from some finite sample (*N* measurements) of the population. The
sample quantities are derived from:

sample mean : ≡

sample variance≡ =

The sample mean and variance approach the true mean and variance as N approaches infinity. But note, especially for small samples, your estimate of the mean and variance may differ from their true (population) values (more below)!

Now we consider what distribution is appropriate for the detection
of photons. The photon distribution can be derived from the *binomial*
distribution, which gives the probability of observing the number,
*x*, of some possible event, given a total number of events *n*, and
the probability of observing the particular event (among all other
possibilities) in any single event, *p*, under the assumption that all events
are independent of each other:

For the binomial distribution, one can derive:

mean≡*xp*(*x*)*dx* = *np*

variance≡*σ*^{2}≡(*x* - *μ*)^{2}*p*(*x*)*dx* = *np*(1 - *p*)

In the case of detecting photons, *n* is the total number of photons
emitted, and *p* is the probability of detecting a photon during our
observation out of the total emitted. We don't know either of these
numbers! However, we do know that *p* < < 1 and we know, or at least we
can estimate, the mean number detected:

In this limit, the binomial distribution asymtotically approaches
the *Poisson* distribution:

From the expressions for the binomial distribution in this limit,
the mean of the distribution is *μ*, and the variance is

variance = [(*x* - *μ*)^{2}*p*(*x*, *μ*)]

variance = *np* = *μ*

Note that the Poisson distribution is generally the appropriate distribution
not only for counting photons, but for *any* sort of counting experiment
where a series of events occurs with a known average rate, and are independent
of time since the last event.

What does the Poisson distribution look like?
Plots
for
*μ* = 2, *μ* = 10, *μ* = 10000.

Note, for large *μ*, the Poisson distribution is well-approximated around
the peak by a *Gaussian*, or *normal* distribution:

This is important because it allows us to use ``simple'' least squares techniques to fit observational data, because these generally assume normally distributed data. However, beware that in the tails of the distribution, and at low mean rates, the Poisson distribution can differ significantly from a Gaussian distribution. In these cases, least-squares may not be appropriate to model observational data; instead, one might need to consider maximum likelihood techniques instead.

The normal distribution is also important because many physical variables
seem to be distributed accordingly. This may not be an accident because
of the *central limit theorem*: if a quantity depends on a number of
independent random variables with ANY distribution, the quantity itself
will be distributed normally (see statistics texts). In observational techniques, we
encounter the normal distribution because one important source of
instrumental noise, *readout noise*, is distributed normally.

You need to understand the expected uncertainties in your observations in order to:

- predict the amount of observing time you'll need to get
uncertainties as small as you need them to do your science,
- answer the question: is scatter in observed data consistent with expected
uncertainties? If the answer is no, they you know you've either learned some
astrophysics or you don't understand something about your observations.
This is especially important in astronomy where objects are faint
and many projects are pushing down into the noise as far as possible.
Really we can usually only answer this probabilistically. Generally,
tests compute the probability that the observations are consistent with
an expected distribution (the null hypothesis). You can then look to see
if this probability is low, and if so, you can reject the null hypothesis.
- interpret your results in the context of a scientific prediction

For example, say we want to know whether some single point is consistent with expectations, e.g., we see a bright point in multiple measurements of a star, and want to know whether the star flared. Say we have a time sequence with known mean and variance, and we obtain a new point, and want to know whether it is consistent with known distribution?

If the form of the probability distribution is known, then you can
calculate the probability of getting a measurement more than some
observed distance from the mean.
In the case where the observed distribution is Gaussian (or approximately
so), this is done using the *error function* (sometimes called
*erf(x)*), which is the integral of a gaussian from some starting
value.

Some simple guidelines to keep in mind follow (the actual situation often
requires more sophisticated statistical tests). First, for
Gaussian distributions, you can calculate that 68% of the points
should fall within plus or minus one sigma from the mean, and 95.3%
between plus or minus two sigma from the mean. Thus, if you have a
time line of photon fluxes for a star, with N observed points, and a
photon noise *σ* on each measurement, you can test whether the
number of points deviating more than 2*σ* from the mean is much
larger than expected. To decide whether any single point is really
significantly different, you might want to use more stringent criterion,
e.g., 5*σ* rather than a 2*σ* criterion; a 5*σ* has much
higher level of significance. On the other hand, if you have far more points
in the range
2 - -4*σ* brighter or fainter than you would expect,
you may also have a significant detection of intensity variations
(provided you really understand your uncertainties on the measurements!).

Also, note that your observed distribution should be consistent with
your uncertainty estimates given the above guidelines. If you have a whole
set of points, that all fall within 1*σ* of each other, something
is wrong with your uncertainty estimates (or perhaps your measurements are
correlated with each other)!

For a series of measurements, one can calculate the *χ*^{2} statistic, and
determine how probable this value is, given the number of points.