1.
(a) Explain how, starting from a sequence of random numbers
distributed uniformly in the interval [0,1), you can generate another sequence of random
numbers that are distributed in the interval [1,4) with density
g(t) = (2/9)(4-t).
(b) Calculate what fraction of these numbers will fall in the interval [1,2).
(c) Use Maple to generate 10,000 such random numbers, and find the
fraction of them that lie in [1,2). Comment on your answer.
(Hint: repeat the experiment (without a restart of Maple).)
Here is some Maple code you could make use of for this. (Suggestion: While you are developing your code, use a very small number of numbers.
Only when you are pretty sure it's working correctly should you do the big run with
10,000 numbers!)
Maple commands useful for this and other work. Information
can be obtained on each of these by typing:
?command rand seq proc if ... then ... else ... end if; for ... from ... to ... do ... end do; nops2. For Extra Credit applied to Exams
(a) Replicate the calculation in the Ringland & Kopecky text
of how the mean and variance evolve for
the case of a stocahstic Birth Process rather than a "Death Process" (radioactive decay).
That is, there is probability &beta &Delta t of the population increasing
by one during a short interval &Delta t, instead of decreasing by one as in the
radioactive decay process.
Here is how you could solve the DE for the variance:
(b) Is the deterministic exponential model
a good idealization of this stochastic process? Answer this
by computing what the coefficient of variation (standard deviation
relative to the mean) does. (E.g. does it have a maximum, or a limit
as t goes to infinity, or does it just keep increasing?)
