OR360
Section 7 Notes
Common Continuous Distributions
1
Exponential Distribution
•
Recall an exponentially distributed random variable
X
has density:
f
X
(
x
) =
λe

λx
,
x
≥
0
,
for some parameter
λ >
0. It has cumulative distribution
F
X
(
x
) = 1

e

λx
. The moment
generating function can be computed:
φ
(
t
) =
E
±
e
tX
²
=
Z
∞
0
e
tx
λe

λx
dx
=
λ
λ

t
Z
∞
0
(
λ

t
)
e

(
λ

t
)
x
dx
=
λ
λ

t
.
•
We have previously shown the memorylessness property:
P
{
X > s
+
t

X > t
}
=
P
{
X > s
}
,
for
s, t
≥
0
.
Example:
In a post oﬃce with two clerks, Alice and Bob are being served, when Charlie
walks in. Charlie’s service will begin when either Alice or Bob leave. Suppose each service
time is exponentially distributed with parameter
λ
. What is the probability that Charlie is
the last to leave?
Solution:
Let
t
be the time that either Alice or Bob leaves (suppose Alice leaves), and let
X
be Bob’s total service time. Then
P
{
X > s
+
t

X > t
}
=
P
{
X > s
}
, so the remaining service
time is still exponentially distributed with parameter
λ
. That is, it’s as if Bob were starting
his service all over again at the time that Charlie’s service starts. Thus, by symmetry, the
probability that Bob ﬁnishes before Charlie is
1
2
.
•
Example:
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 Summer '08
 WEBER
 Normal Distribution, Probability theory, moment generating function

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