"""
Illustrate the connections bettwen the uniform, exponential, gamma and
normal distributions by simulating waiting times from a radioactive
source using the random number generator. Verify the numerical
results by plotting the analytical density functions from scipy.stats
"""
import numpy
import scipy.stats
from pylab import figure, show, close
# N samples from a uniform distribution on the unit interval. Create
# a uniform distribution from scipy.stats.uniform and use the "rvs"
# method to generate N uniform random variates
N = 100000
uniform = XXX # the frozed uniform distribution
uninse = XXX # the random variates
# in each time interval, the probability of an emission
rate = 20. # the emission rate in Hz
dx = 0.001 # the sampling interval in seconds
t = numpy.arange(N)*dx # the time vector
# the probability of an emission is proportionate to the rate and the interval
emit_times = XXX
# the difference in the emission times is the wait time
wait_times = XXX
# plot the distribution of waiting times and the expected exponential
# density function lambda exp( lambda wt) where lambda is the rate
# constant and wt is the wait time; compare the result of the analytic
# function with that provided by scipy.stats.exponential.pdf; note
# that the scipy.stats.expon "scale" parameter is inverse rate
# 1/lambda. Plot all three on the same graph and make a legend.
# Decorate your graphs with an xlabel, ylabel and title
fig = figure()
ax = fig.add_subplot(111)
p, bins, patches = XXX # use ax.hist
l1, = ax.plot(bins, XXX, lw=2, color='red') # the analytic result
l2, = ax.plot(bins, XXX, # use scipy.stats.expon.pdf
lw=2, ls='--', color='green')
ax.set_xlabel('waiting time')
ax.set_ylabel('PDF')
ax.set_title('waiting time density of a %dHz Poisson emitter'%rate)
ax.legend(XXX, XXX) # create the proper legend
# plot the distribution of waiting times for two events; the
# distribution of waiting times for N events should equal a N-th order
# gamma distribution (the exponential distribution is a 1st order
# gamma distribution. Use scipy.stats.gamma to compare the fits.
# Hint: you can stride your emission times array to get every 2nd
# emission
XXX
# plot the distribution of waiting times for 10 events; again the
# distribution will be a 10th order gamma distribution so plot that
# along with the empirical density. The central limit thm says that
# as we add N indenpendent samples from a distribution, the resultant
# distribution should approach the normal distribution. The mean of
# the normal should be N times the mean of the underlying and the
# variance of the normal should be 10 times the variance of the
# underlying. HINT: Use scipy.stats.expon.stats to get the mean and
# variance of the underlying distribution. Use scipy.stats.norm to
# get the normal distribution. Note that the scale parameter of the
# normal is the standard deviation which is the square root of the
# variance
XXX
show()
