ASSIGNMENT 9
There are two parts to this assignment. The first part is on WeBWorK — the link is available on the course
webpage. The second part consists of the questions on this page. You are expected to provide full solutions
with complete arguments and justifications. You will be graded on the correctness, clarity and elegance of
your solutions. Your answers must be typed or very neatly written. They must be stapled, with your name
and student number at the top of each page.
This assignment is due at 10:00 a.m. on Friday, January 24. The written assignment is to be handed
in by that time. The online assignment will also close at that time.
1. This question refers to Figure 1 and equation (2) in the 2007 paper “Use of the gamma distribution to
represent monthly rainfall in Africa for drought monitoring applications” in the International Journal of
Climatology. The paper is available on the course webpage.
(a) The curves in Figure 1 represent probability distributions described by functions f (x) of the form
given in equation (2). Suppose we fix the shape parameter α and scale parameter β, and let F (x)
be the area under the graph of f (t) from t = 0 to t = x. Sketch the graph of F (x), indicating and
justifying any critical points, inflection points or asymptotes.
(b) What does F (x) represent in terms of rainfall accumulation?
(c) Why would an accurate model F (x) be helpful? Explain your answer in one or two paragraphs. Your
answer should not be about mathematics per se. It may include, but is not limited to, topics in
agriculture, ecology, health, atmospheric science and international relations.
b
Z
2. Suppose f is continuous and
f (t) dt = 0. Prove that there exists a point c such that f (c) = 0.
a
Z
3. (a) Prove that
a
Z
0
a
f (a − x) dx.
f (x) dx =
0
Z
(b) Prove, using part (a) or otherwise, that
0
π
π2
x sin x
dx
=
.
1 + cos2 x
4