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AAE 706
Homework #1
I-
Think of a fixed site outside the building in which you are at this moment. Let X be the
temperature at that site at noon tomorrow. Choose a number x1 such that
Pr (X < x1) = Pr (X > x1) = 1/2.
Next choose a number x2 such that
Pr (X < x2) = Pr (x2 < X < x1) = 1/4.
Finally, choose numbers x3 and x4 (x3 < x1 < x4) such that
Pr (X < x3) + Pr (X > x4) = Pr (x3 < X < x1) = Pr (x1 < X < x4) = 1/3.
a/
Using the values of x1 and x2 that you have chosen and a table of the standard normal distribution,
find the unique normal distribution for X that satisfies your answers (x1, x2).
b/
Assuming that X has the normal distribution established in a/, find from the tables the values
which x3 and x4 must have. Compare these values with the values you have chosen. Decide
whether or not your distribution for X can be represented approximately by a normal distribution.
II -
The joint probability function of the random variables X and Y is represented by the following
table:
X
Y
5
6
7
8
1
2
3
.01
.06
.02
.18
.09
.03
.24
.12
.04
.06
.03
.12
a/
b/
c/
d/
e/
f/
Determine the marginal probability functions of X and Y.
Are X and Y independent?
What is the conditional probability function of X, given Y = 7?
What is the expected value of Y, given X = 3?
What is the expected value of Y? of X?
What is the variance of X? The variance of Y? The covariance between X and Y? The
correlation between X and Y?
III -
You face a decision problem involving three states of nature with prior probabilities Pr (a1) = .15,
Pr (a2) = .30 and Pr (a3) = .55. To gain further information, you consult an expert who gives you a
forecast (z) with conditional probabilities:
Pr (z a1) = .3; Pr (z a2) = .50; Pr (z a3) = .10
If you are a Bayesian learner, what probabilities do you want to use in your decision?