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14.30 Introduction to Statistical Methods in Economics
Spring 2009
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#3
14.30

Intro. to Statistical Methods in Economics
Instructor: Konrad Menzel
Due: Tuesday, March
3,
2009
Question
One
1. Write down the definition of a cumulative distribution function (CDF). Explain what
it means in words, perhaps using an example.
Solution to
(1): One definition of the CDF is
f
(.) :
R
H
[0,1] where
f (x)

PT(X
5
x). The CDF tells us the cumulative probability up to particular point
of the ordered support of the random variable, X. What this means is that we
can know what the chances are that something less than or equal to (or to the
left, depending on how you wish to interpret the ordering) an outcome,
x, occurs.
2. Verify whether the following function is a valid CDF. If yes, draw a graph of the
x
F
x
(x)
1
3
4
1
2
1
5
1
2
3
4
5


Solution to (2): The function is in fact a valid CDF. It is bounded below by
zero and above by one. It also satisfies the left and right limit conditions as
lim,,,
Fx(x)
=
0 and lim,,,
=
1. However, it is a mixture random
variable where it has a continuous distribution and a mass point at 2. The PDF
by MIT OpenCourseWare.
Image
is the following equation:
3.
Verify that the following function is a valid PDF and draw the corresponding CDF
a
Solution to
(3):
This function is, in fact, a valid PDF. It is po
F
x
(x)
1
2
3
4
5
6
x
1
3
sitive everywhere
and integrates to
1 (the triangle
Image by MIT OpenCourseWare.
has area of
and the interv
$
al from 5 to
6 has
area of
which together sum to
1).The CDF is straightforward. I will write the
PDF and then CDF down analytically first, to make for easier integration:
Drawing this curve is relatively straightforward, at least if you pay little attention
to detail as I am not a graphic designer:
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View Full Document Question
Two
1. Give a p.d.f. whose c.d.f. is not continuous but is rightcontinuous.
Solution to (1): This will have to come from a distribution with at least one mass
point (or it could be a completely discrete distribution). Konrad's lecture notes
have an example of the CDF:
0
1
1
3
2
3
0
1
2
3
4
5
6
F(x)
F(x)
x
1
1
2
1
6
Image by MIT OpenCourseWare.
Image by MIT OpenCourseWare.
True/false/uncertain: Always give a brief explanation if the statement is true, or counter
examples and a short explanation of the counterexamples if the statement is false or uncer
tain.
1. If P(A1B)
>
P(A) and P(A1C)
>
P(A), then P(AIB,C)
>
P(A).
Solution to (1): False. Just because two conditional probabilities are large, does
not mean that their joint probability will not be large. One example is the prob
ability of getting sick (event A), given it is winter time (event
B). You are more
likely to get sick during winter than the average during the year. You are also
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This note was uploaded on 01/30/2010 for the course STAT 430 taught by Professor Jones during the Fall '10 term at Napa Valley College.
 Fall '10
 jones

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