Posted: September 13th, 2017

Mathematics Statistics

Mathematics Statistics

Project description

1. Let
X
be a continuous random variable with distribution function
F
and pdf
f
given by
F
(
x
) =
1
2
+
1
arctan (
x
) and
f
(
x
) =
1
(1 +
x
2
)
(
x
2
R
)
:
Determine the distribution function and the pdf of
Y
=
aX
+
b
, where
a
and
b
are
xed constants,
a
being strictly positive. Determine the pdf of
Z
= log(
j
X
j
).
1
[5]
2. Let
Y
= 2(
X
1)
2
1, where
X
is uniformly distributed over the interval [0
;
2].
Determine the pdf of
Y
and the expected value of
Y
.
[5]
3. Let
Y
1
=
X
2
1
and
Y
2
=
X
1
X
2
, where
X
1
and
X
2
are two jointly continuous random
variables with joint pdf
f
(
x
1
;x
2
) =
(
2
x
1
if 0
< x
1
<
1
;
0
< x
2
<
1
0 otherwise
:
Determine the joint pdf of
Y
1
and
Y
2
. Identify the marginal distribution of
Y
1
and
determine the marginal pdf
Y
2
. Are
Y
1
and
Y
2
independent? (Explain.)
[10]
Total
[20]
Postgraduate problem
. Let
X
and
Y
be independent standard Cauchy random variables.
Determine the pdf of
aX
+
bY
, where
a;b
6
= 0.
1
This distribution is called the
hyperbolic secant distribution
. Can you see why?

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