Problem 12   Due 2/19/01

The decay of radioactive isotopes is governed
by the Poisson distribution.  In this problem
we examine both the mean and the variance of
this distribution.  Suppose we have a sample of
radioactive material.  If we observe the material
for some time t we will get, on average, r*t decay
events, where r is the decay rate of the material.
The probability of observing k events during this
period is:

p(k) = (r*t)^k / k!  exp(-r*t)

a). Write a script to calculate the mean and the variance
of this distribution for r*t = 5 and r*t = 500.
(Hint: you may wish to calculate the log of p for larger
values of r*t first)

b). On two plots (one for each value of r*t) graphically
compare the poisson distribution to a gaussian distribution
with the same mean and variance.  You should find that the
two collapse for the larger value of r*t.

c). If r = 30 counts per minute (the true value), how long
do we have to observe the decay to capture this value to
within 10% accuracy at the 95% confidence level?
(This sort of question is important in the biomedical sciences
in which radioactive tracers are used to determine the way
in which drugs are taken up and distributed throughout tissues.
Radioactive tracers are often used in animal tests, for
example)