Math 2260 HW #1
Due 10:10 AM Friday, January 11
Reading: Hass §5.1–5.5
Problems: Do the assignment “HW1” on WebWork. In addition, write up solutions to the
following problems and hand in your solutions in class on Friday.
1. Consider the curves y = x1 , y =
1
,
x2
and x = 6.
(a) Sketch the region enclosed by these curves.
(b) Find the exact value (i.e., not a decimal approximation) of the area of this region.
2. In general, if T is a random variate, then X obeys some probability distribution which says
where T is likely to be. The most famous example is the so-called normal or Gaussian
distribution. A random variate has a certain probability of taking on values in an interval
which is given by probability density function. Specifically, if the random variable T is chosen
from a probability distribution with probability density function f (x), then the probability
that T lies between the real numbers a and b (with a < b) is given by
Z
b
Pr[a ≤ T ≤ b] =
f (x) dx.
a
Pictorially, this means that the area under the graph of the probability density function
between x = a and x = b is precisely the probability that T is between a and b.
Now, suppose T is a random variate distributed according to the Kuramaswamy(2,7) distribution, which has the probability density function f (x) = 14x(1 − x2 )6 graphed below:
2.0
1.5
1.0
0.5
0.2
0.4
What is the probability that T is between
0.6
1
4
1
and 21 ?
0.8
1.0