MATH 11300
Probability
AUTUMN 2009
Problem Sheet 9
Questions marked * are to be handed in.
*1. Let
X
be a continuous random variable taking values in the interval [
-
3
,
3], with probability
density function (pdf)
f
X
given by
f
X
(
x
) =
(9
-
x
2
)
/
36
-
3
≤
x
≤
3
0
otherwise
.
(a) Find the distribution function
F
X
for
X
.
(b) Find
P
(
X <
0)
, P
(
-
2
≤
X
≤
1) and
P
(
X >
2
.
5).
2. Let
X
be a continuous random variable with distribution function
F
X
given by
F
X
(
x
) =
1
-
(1 +
x
)
e
-
x
x >
0
0
otherwise
.
Find the density function
f
X
for
X
. Hence find
E
(
X
) and Var(
X
)
*3. Let
Y
be a continuous random variable taking values in the interval [0
,
1]. Assume
Y
has
probability density function (pdf) proportional to
y
(1
-
y
), i.e. there exists a fixed constant
c
such that the pdf
f
Y
is given by
f
Y
(
y
) =
cy
(1
-
y
)
0
≤
y
≤
1
0
otherwise.
(a) Find the value
c
must take for this to be a pdf.
(b) Sketch the pdf
f
Y
.
(c) Find and sketch the distribution function
F
Y
for
Y
.
*4. Let
Y
=
X
2
where
X
∼
U
(
-
1
,
2). Find the distribution function
F
Y
for
Y
. Hence find
the density function for
Y
. [Hint: For
F
Y
(
y
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- Spring '09
- Probability, Probability distribution, Probability theory, probability density function, density function, distribution function Fx
-
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