\section{Statistical distributions}
\label{sec:stats_distributions}
We explore a handful of the statistical distributions in
\texttt{scipy.stats} module and the connections between them. The
organization of the distribution functions in \texttt{scipy.stats} is
quite elegant, with each distribution providing random variates
(\texttt{rvs}), analytical moments (mean, variance, skew, kurtosis),
analytic density (\texttt{pdf}, \texttt{cdf}) and survival functions
(\texttt{sf}, \texttt{isf}) (where available) and tools for fitting
empirical distributions to the analytic distributions (\texttt{fit}).
in the exercise below, we will simulate a radioactive particle
emitter, and look at the empirical distribution of waiting times
compared with the expected analytical distributions. Our radioative
particle emitter has an equal likelihood of emitting a particle in any
equal time interval, and emits particles at a rate of 20~Hz. We will
discretely sample time at a high frequency, and record a 1 of a
particle is emitted and a 0 otherwise, and then look at the
distribution of waiting times between emissions. The probability of a
particle emission in one of our sample intervals (assumed to be very
small compared to the average interval between emissions) is
proportional to the rate and the sample interval $\Delta t$, ie
$p(\Delta t) = \alpha \Delta t$ where $\alpha$ is the emission rate in
particles per second.
\begin{lstlisting}
# a uniform distribution [0..1]
In [62]: uniform = scipy.stats.uniform()
# our sample interval in seconds
In [63]: deltat = 0.001
# the emission rate, 20Hz
In [65]: alpha = 20
# 1000 random numbers
In [66]: rvs = uniform.rvs(1000)
# a look at the 1st 20 random variates
In [67]: rvs[:20]
Out[67]:
array([ 0.71167172, 0.01723161, 0.25849255, 0.00599207, 0.58656146,
0.12765225, 0.17898621, 0.77724693, 0.18042977, 0.91935639,
0.97659579, 0.59045477, 0.94730366, 0.00764026, 0.12153159,
0.82286929, 0.18990484, 0.34608396, 0.63931108, 0.57199175])
# we simulate an emission when the random number is less than
# p(Delta t) = alpha * deltat
In [84]: emit = rvs < (alpha * deltat)
# there were 3 emissions in the first 20 observations
In [85]: emit[:20]
Out[85]:
array([False, True, False, True, False, False, False, False, False,
False, False, False, False, True, False, False, False, False,
False, False], dtype=bool)
\end{lstlisting}
The waiting times between the emissions should follow an exponential
distribution (see \texttt{scipy.stats.expon}) with a mean of
$1/\alpha$. In the exercise below, you will generate a long array of
emissions, compute the waiting times between emissions, between 2
emissions, and between 10 emissions. These should approach an 1st
order gamma (aka exponential) distribution, 2nd order gamma, and 10th
order gamma (see \texttt{scipy.stats.gamma}). Use the probability
density functions for these distributions in \texttt{scipy.stats} to
compare your simulated distributions and moments with the analytic
versions provided by \texttt{scipy.stats}. With 10 waiting times, we
should be approaching a normal distribution since we are summing 10
waiting times and under the central limit theorem the sum of
independent samples from a finite variance process approaches the
normal distribution (see \texttt{scipy.stats.norm}). In the final
part of the exercise below, you will be asked to approximate the 10th
order gamma distribution with a normal distribution. The results
should look something like those in
Figure~\ref{fig:stats_distributions}.
\lstinputlisting[label=code:stats_distributions,caption={IGNORED}]{problems/stats_distributions.py}
\begin{figure}
\begin{centering}\includegraphics[width=4in]{fig/stats_distributions}\par\end{centering}
\caption{\label{fig:stats_distributions}}
\end{figure}
