STA 3007 Applied Probability
2005
Tutorial 9
1. Poisson Process
(a) Distribution Associated with the Poisson Process
i. Example
1 Let
X
1
(
t
) and
X
2
(
t
) be independent Poisson processes having parameters
λ
1
and
λ
2
,
respectively. What is the probability that
X
1
(
t
) = 1 before
X
2
(
t
) = 1 ?
(b) The Uniform Distribution and Poisoon Processes
i. Theorem 3.6 Let
W
1
, W
2
, . . .
be the occurrence times in a Poisson process of rate
λ >
0.
Conditioned on
X
(
t
) =
n
, the random variables
W
1
, W
2
, . . . , W
n
have the joint probability
density function
f
W
1
,...,W
n

X
(
t
)=
n
(
w
1
, . . . , w
n
) =
n
!
t

n
for
0
< w
1
<
· · ·
< w
n
≤
t.
ii. Example
1 (P.301P.303) Viewing a fixed mass of a certain radioactive material, suppose that
alpha particles appear in time according to a Poisson process of intensity
λ
. Each par
ticle exists for a random duration and is then annihilated. Suppose that the successive
lifetimes
Y
1
, Y
2
, . . .
of distinct particles are independent random variables having the
common distribution function
G
(
y
) = Pr
{
Y
k
≤
y
}
.
Let
M
(
t
) count the number of
alpha particles existing at time
t
. What is the probability distribution of
M
(
t
) under
the condition that
M
(0) = 0.
1
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Let
X
(
t
) be the number of particles created up to time
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 Spring '11
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 Normal Distribution, Probability, Probability theory, probability density function, Poisson process, Cumulative distribution function

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