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A solution to a probability homework problem involving the lifetime of chips, represented as a sequence of bernoulli trials. Calculations and matrix notation to determine the probability of a chip's lifetime exceeding a certain threshold and the expected number of trials required. It also discusses the concept of eigenvalues and eigenvectors in relation to the problem.
Typology: Exercises
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k k
k
p p p
k
p p p p p
− −
=
−
For an individual chip, the problem that the lifetime exceeds
seconds is:
1
1
P lifetime e e p
α
α
− ⋅
−
If we consider the testing of whether each chip lifetime exceeds
as a sequence of
Bennoulli trials, then
10
1 1 10
5
k k
k
P k e e
k
− − −
=
(a)
0 1
p = p =
(b)
0 0 1
1 0 1
p n p n p n
p n p n p n
In matrix notation, we have
0 1 0 1
p n p n p n p n
, or p ( n + 1) = p n ( ) Ρ
2
(c)
p
p
p
p
p
(d)
2
p
p p
p p p p
in general ( ) (0)
n
p n = p P. To find
n
P , we note that if P has eigenvalues
1
2
and eigenecotrs
1
e ,
2
e , then
1 1
1 2
2
, where [ , ]
e e
−
Λ
, and
1 1 1
1 -1 1
1
( )( ) ( ) times
n
n
n
− − −
− −
−