1
Probability Homework Solution #4
2.71
2100 3
0
100 99 98 2
4
[3 or more errors] 1 [2 or fewer errors]
100
1(1)10
100 99
1 (1 ) 100(1 ) (1 )
2
1 099985 1.5 10
kk
k
PP
pp p
k
p
pp pp
−−
=
−
=−
⎛⎞
=− − ⇐ =
⎜⎟
⎝⎠
×
⎧⎫
=− − + − + −
⎨⎬
⎩⎭
=− = ×
∑
2.75
For an individual chip, the problem that the lifetime exceeds 1
α
seconds is:
11
1
lifetimePeep
αα
α
−⋅ −
⎡⎤
>= =
⎢⎥
⎣⎦
If we consider the testing of whether each chip lifetime exceeds 1
α
as a sequence of
Bennoulli trials, then
[]
10 1110
5
10
5 ( ) (1 ) 0.289
kk
k
Pk e e
k
−−−
=
⎛⎞
≥= − =
⎜⎟
⎝⎠
∑
2.82
(a)
01
11
(1) , (1)
22
pp==
(b)
001
101
21
(1) () ()
36
15
(1) () ()
36
p
npnpn
p
npnpn
+= +
+= +
In matrix notation, we have
[][]
01 01
21
33
(1),(1) (),()
15
66
pn pn pnpn
⎡
⎤
⎢
⎥
++=
⎢
⎥
⎢
⎥
⎢
⎥
⎣
⎦
, or (1) ()
p
npn
+
=Ρ
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