Jacobs University Bremen
School of Engineering and Science
Peter Oswald
Fall Term 2010
120201 ESM3A — Problem Set 8
Issued: 4.11.2010
Due: Wednesday 10.11.2010 (in class)
This is about continuous random variables and their properties. Compare e.g. Lectures 10
and 11 in the posted script.
8.1.
Which of the following are distribution functions? For those that are, give the correspond
ing density function.
a)
F
(
x
) =
(
e

1
/x
if
x >
0,
0
otherwise,
b)
F
(
x
) =
e
x
e
x
+
e

x
, x
∈
IR,
c)
F
(
x
) =
e

x
2
+
e
x
e
x
+
e

x
, x
∈
IR.
8.2.
The median med(
X
) of a continuous random variable
X
is the real number
t
that satisfies
P
(
X
≤
t
) = 1
/
2 (it can happen that there is a whole interval of
t
, in which case the
median is usually defined as the midpoint of this interval). In contrast to expectations,
the median always exists.
Find the median of a uniform distribution on [
a, b
], of a normally distributed random
variable
N
(
μ, σ
2
), of an exponentially distributed random variable (parameter
λ
), of a
Cauchy random variable, and of the Pareto random variable (see next problem).
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 Spring '11
 Prof. Dr. Peter Oswald
 Normal Distribution, Standard Deviation, Variance, Probability theory, probability density function

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