Here, we shall use the Cray random number generator, ranf(), to
perform emissions according to
Figure 10
and Equations
(23) and (24).
We will do this
by generating vectors (in the computer sense) of random numbers---one
vector for 
and one vector for 
. Using these, we'll generate one
vector of 
values and one other vector of 
values. As we
generate ``many'' emissions, we should expect the distribution of emissions
to approach: (1) a uniform distribution in 
and (2) 
in 
. First, generate 10,000 emissions
(requiring 20,000 random numbers - 10,000 to obtain the 10,000 values of
, and another 10,000 to obtain the 10,000 values of 
).
Partition these emissions into groups of N = 10, 100, 1,000 and 10,000
(you can do this by referencing the first 10, 100, ... elements in your
and 
arrays).
(a) Consider the distribution function for 
.
Partition the domain into 36 intervals (bins) of 
each
(
--
, 
--
, ...).
For each N, plot the function of emissions which fall into the bins vs. the
bin centerpoint (i.e., plot 
vs.
, 
, ...).
Comment on the ``convergence'' of the distributions.
(b) Now perform a similar test for 
.
Divide 
into 9 bins of 
each, with
, 
, ....
Plot 
vs. 
for all 5 N's.
Comment on the ``convergence'' here, too.
Notes:
1. The exact answers are:
(a)
(b)
2. Do the above calculations as efficiently as possible (if on the Cray, use hpm to assess MFLOP's).
Hand in: (1) your program, (2) the plots, and (3) discussions.
