Spring 2010
Pstat160b
HO #3
Exponential Distribution summary
Notation:
We denote the exponential distribution with parameter
λ >
0 by
E
(
λ
).
Interpretation:
Exponential random variables will often be interpreted as the time an event occur, or as a ’random
clock’.
Basic properties
:
p.d.f
f
(
x
) =
λe

λx
x
≥
0
C.d.f
F
(
x
) = 1

e

λx
x
≥
0
Tail probability
P
(
X
≥
x
) = 1

F
(
x
) =
e

λx
x
≥
0
M.G.F.
M
(
t
) =
λ
λ

t
t
6
=
λ
mean
E
(
X
) =
1
λ
variance Var(
X
) =
1
λ
2
Further properties
:
•
Memoryless property:
P
(
X > x
+
y

X > y
) =
P
(
X > x
)
x,y >
0
.
•
Hazard rate (or juste “rate”): For a continuous random variable
T
, the probability that
T
occurs between
t
and
t
+
h
, given it doesn’t occur before
t
is given by the density
r
(
t
) called hazard rate function which is
r
(
t
) =
f
(
t
)
1

F
(
t
)
r
(
t
) =
λ
for exponential
E
(
λ
)only
So we can interpret the parameter
λ
as the “inﬁnitesimal probability that something is going to happen next” or
the clock to ring, or the rate at which it happens (say customer arrival/unit time), that is
P
(
X
∈
[0
,
0 +
dx
]) =
P
(
X
∈
[
x,x
+
dx
]

X > x
) =
λdx
(note how the memoryless property relates to constant hazard rate).
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '10
 bonnet
 Probability theory, Hazard rate function, hazard rate, exponential random variables, Exponential Distribution summary, λn e−λx

Click to edit the document details