1
Computer Assignment #3
Parameter Estimation and Functions of a Random Variable
Purpose
Thus far, we have dealt with the problem of characterizing the behavior of a random variable.
The
next logical problems to be solved are how, given a set of observed data, to estimate the parame-
ters of the distribution that governs the observed phenomenon, and how to characterize the behav-
ior of another functionally related random variable.
Both of these have been addressed in the
textbook homework, but a computer assignment allows us to explore larger data sets and more
complex interrelations than are easily done by hand.
This assignment will provide you with a
fairly simple problem to give you a feel for estimating probabilistic models and the effects of sam-
ple size on estimation accuracy, as well as the relationship between two random variables — per-
haps more of a feeling than may be attainable through a formal mathematical treatment.
Additionally, you will also gain some experience writing M
ATLAB
functions and solving a (non-
linear) algebraic equation using M
ATLAB
.
The central problem in this assignment is the probabilistic modeling of the wind velocity at the
location of a windmill and the power output by the windmill.
First, we will set up the problem
and give some theoretical background, then study the parameter estimates and con
f
dence inter-
vals from multiple sets of samples of the wind velocity, apply some of our analytical knowledge in
f
nding the CDF of windmill power output, and compare that to generated samples of the power
output.
Theory
Let us begin with some basic empirical relations you will need which have been determined by
researchers in the
f
eld of wind energy.
W
ind
V
elocity
:
It has been determined that wind velocity in a given area may be modeled by a
random variable
, in units of m/s, having the following PDF and CDF:
1
,
,
,
,
W
indmill P
ower
:
The relation between wind velocity
and power output
is somewhat com-
plex because of the design of the windmill.
(
a
) If the wind velocity is less than 4 m/s, there is
insuf
f
cient wind force to overcome the friction in the components of the windmill; consequently,
for wind velocities less than 4 m/s, the power output is zero.
(
b
) At high wind velocities, the
windmill could be damaged through extended periods at high power outputs; thus, the windmill
blades are designed to “feather” (
i.e.
, change their angle of attack to the oncoming wind) so as to
operate at a constant power level for all wind velocities at or above 13 m/s.
(
c
) For intermediate
wind velocities (
i.e.
,
), the power output is proportional to the cube of the
wind velocity in m/s
,
1. Note that the mean and variance are
and
,
where
is the Gamma function.
In MATLAB,
may be computed with
gamma(x)
.