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ESO 209: PROBABILITY & STATISTICS
Semester 2: 200708
Assignment #4
Instructor: Amit Mitra
[1]
Find the expected number of throws of a fair die required to obtain a 6.
[2]
Consider a sequence of independent coin flips, each of which has a probability
p
of
being heads.
Define a random variable
as the length of the run (of either heads or
tails) started by the first trial.
Find
X
( )
E X
.
[3]
Find
( )
E X
(if it exists) in the following cases:
(a)
has the p.m.f.
X
()
1
1
if
1,2,.
...
0o
t
h
e
r
w
xx
x
PX x
−
⎧
+=
⎪
==
⎨
⎪
⎩
i
s
e
.
(b)
has the p.d.f.
X
( )
2
12
i
f
1
t
h
e
r
w
i
s
fx
⎧
>
⎪
=
⎨
⎪
⎩
e
.
(c)
has the p.d.f.
X
2
11
,.
1
x
x
π
=−
∞
<
+
<
∞
[4]
Find the mean and variance of the following distributions
(a)
( )
1
,0
1
,
0
a
fx a
x
x
a
−
=<
<
>
(b)
1
,
1,2,.
.., ;
0
x
nn
n
>
an integer
(c)
2
3
1,
0
2
2
x
x
=− <
<
[5]
Find the mean and variance of the Weibull random variable having the p.d.f.
1
exp
if
0
otherwise.
cc
cx
x
x
aa
a
µµ
µ
−
⎧
⎧⎫
−−
⎪⎪
⎛⎞
⎪
−>
⎨⎬
⎜⎟
=
⎝⎠
⎨
⎩⎭
⎪
⎩
Where,
and
0,
0
ca
>>
( )
∈−∞∞
[6]
A median of a distribution is a value
such that
m
( )
PX m
≤≥
and
≥≥
.
Find
the
median
of
the
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 Spring '08
 j.john
 Microelectronics

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