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.