| 20
February |
- Complete textbook exercises 4.1, 4.6, 4.7,
4.8, 4.9, 4.10, 4.11, 4.12, 4.13,
4.14, 4.15, 4.16, 4.17, 4.18, and
4.20. (90 min.)
- Compute by hand, without reference to a table,
calculator, or computer, the natural logarithms of the integers 1,
2, 3, ..., 20 to two decimal places. Then check your
answers. (Excel will compute them for you using its LN()
function.) If you got any incorrect--even by one in the last
digit--identify the reason why and redo the computation until you
can get them all correct. Then draw a precise graph of y =
ln(x) for x between 0.05 and 20. (45 min.)
- Compute by hand, to two decimal places, the
logarithms of 9.5, 9.6, 9.7, ..., 10.5. (1 minute.)
- * A risk assessor assumes, as part of a modeling
exercise, that the dose of a carcinogen received by a human through
contact with it in the environment is proportional to the product of
its concentration and the duration of the exposure. Suppose
she has data suggesting the concentration is lognormally distributed
throughout the environment with a mean of 37 parts per million (ppm)
and a standard deviation of 24 ppm, and the exposure duration in the
affected human population also is lognormally distributed with a
mean of 5 years and standard deviation of 2 years. Assume the
exposure duration is independent of the concentration (why is this
usually a reasonable assumption? Why should it be
checked?). A "deterministic" risk model might
well use these mean values for its inputs, resulting in 37 * 5 = 185
ppm-years. What is the probability that the product of
concentration and duration will exceed 185
ppm-years? What is the probability that this
product will exceed 1,000 ppm-years? For what value is this
"exceedance" probability exactly equal to five
percent? (Hint: work in terms of the logarithms of
concentration and exposure and recall that the sum of two normal
distributions is normal.) (30 min.)
- Using a table of the cdf of the standard normal
distribution (or an equivalent table, such as the one handed out in
class), find the following probabilities, or determine why they do
not make sense. Z is a random variable distributed normally
with mean 0 and standard deviation 1.
a. Z is greater than 2.
b. Z is between -0.5 and 0.5.
c. Z is greater than 1 or less than -0.5.
d. Z is not equal to 0.
e. Z2 is greater than 4.
f. exp(Z) is greater than 1.
(15 min.)
- Graph the cdf of the following distributions.
a. A uniform random variable with values between -1 and 1.
b. A uniform random variable with values between 0 and 3.
c. A mixture of 25% (a) and 75% (b).
d. A variable that has a 25% chance of being 0, a 50% chance
of being 1, and a 25% chance of being 2.
e. A mixture of 50% (a) and 50% (d).
(30 min.)
- Read Chapter 5, pages 201 through 228. (30
min.)
- To prepare for the quiz, review last week's
material on Excel, simulations, and solving probability problems
using box models.
Total estimated time: 4:01 hours + review time.
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