NTHU MATH 2820, 2008, Lecture Notes
Ch1~6, p.73
explanations.
1. if
n
large, the pmf of
B
(
n, p
) is not easily calculated. Then, we can
approximate them by pmf of
P
(
λ
), where
λ
=
np
.
2. Let
X
be the number of times some event occurs in a given time interval
I
. Divide the interval into many small subintervals
I
k
,
k
=1
,...,n
,o
f
equal length. Let
N
k
be the number of events occurring in
I
k
.Wh
en
we can assume
N
1
,...,N
n
are independent and approximately
∼
B
(
p
),
X
has a distribution near
P
(
λ
), where
λ
=
.
•
pmf:
p
(
x
)=
λ
x
x
!
e
−
λ
,x
=0
,
1
,
2
,...
0
,
otherwise
.
•
mgf:
e
λ
(
e
t
−
1)
,t
∈
R
.
•
mean:
λ
•
variance:
λ
•
parameter:
λ
>
0
•
example:
number of phone calls coming into an exchange
during a unit of time
made by Shao-Wei Cheng (NTHU, Taiwan)
Ch1~6, p.74
Note:
Let
X
i
∼
P
(
λ
i
),
i
,...,k
,and
X
1
,...,X
k
are independent. Then,
Y
=
X
1
+
···
+
X
k
∼
P
(
λ
1
+
+
λ
k
).
Definition 4.8
(Hypergeometric distribution
HG(r, n, m)
, sec 2.1.4)
Suppose that an urn contains
n
black balls and
m
white balls.
Let
X
denote the number of black balls drawn when taking
r
balls without replacement. Then,
X
follows hypergeometric
distribution.
•
pmf:
p
(
x
n
x
m
r
−
x
n
+
m
r
,
x
,
1
,...,
min(
r, n
)
,
r
−
x
≤
m
0
,
otherwise