ESO 209: PROBABILITY & STATISTICS
Semester 2: 2007-08
Assignment #5
Instructor: Amit Mitra
[1]
Let
be a Poisson random variable with parameter
X
λ
.
Find the probability mass
function of
.
2
5
Y
X
=
−
[2]
Let
be Binomial random variable with parameters
and
X
n
p
.
Find the
probability mass function of
Y
n
X
=
−
.
[3]
Consider the discrete random variable
with the probability mass function
X
(
)
(
)
(
)
(
)
(
)
(
)
1
1
2
,
1
,
0
,
5
6
1
10
1
,
2
,
3
15
30
30
P X
P X
P X
P X
P X
P X
= −
=
= −
=
=
=
=
=
=
=
=
=
1
5
1
.
Find the probability mass function of
2
.
Y
X
=
[4]
The probability mass function of the random variable
is given by
X
(
)
1
2
0,1,2,...
3
3
0
otherwi
x
x
P X
x
⎧
⎛
⎞
=
⎪
⎜
⎟
=
=
⎨
⎝
⎠
⎪
⎩
se.
Find the distribution of
(
)
1 .
Y
X
X
=
+
[5]
The probability density function of the random variable
is
X
(
)
1
0
1
0
otherwise.
X
x
f
x
<
<
⎧
=
⎨
⎩
i.e.
(
)
~
0,1
X
U
.
Find the distribution of the following functions of
X
(a)
Y
X
=
(b)
2
Y
X
=
(c)
2
3
Y
X
=
+
(d)
log
;
0.
Y
X
λ
λ
= −
>
[6]
Let
be a random variable with
X
(
)
0,
,
U
θ
0
θ
>
distribution. Find the distribution
of
(
)
min
,
2 .
Y
X
θ
=

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