ORIE 361/523 – Homework 6
Instructor: Mark E. Lewis
due March 13, 2008 (drop box)
1. A machine has two critically important parts and is subject to 3 different types of shocks.
Shocks of type i occur at times of a Poisson process with rate
λ
i
. Shocks of types 1 break
part 1, those of type 2 break part 2 while those of type 3 break both parts. Let
U
and
V
be
the failure times of the two parts.
(a) Find
P
[
U > s, V > t
]
.
(b) Let
t
→
0 and
s
→
0 in the previous part to show that
U
and
V
each have exponential
distributions.
(c) Are
U
and
V
independent?
2. Let
{
N
(
t
)
, t
≥
0
}
be a Poisson process with rate
λ
, that is independent of the nonnegative
random variable
T
with mean
μ
and variance
σ
2
. Find
(a) Cov(
T, N
(
T
))
(b) Var(
N
(
T
))
3. Consider
n
components with independent lifetimes which are such that component
i
functions
for an exponential time with rate
λ
i
. Suppose that all components are initially in use and
remain so until they fail.
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 Spring '07
 LEWIS,M.
 Probability theory, Exponential distribution, Poisson process, new kidney, Mark E. Lewis

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