Fall 2008
Final Exam
Page: 1 of 3
Introduction to Probability
Monday, December 16, 2008
STA 4321/5325
Instructions:
Please turn of your cell phones. Please write
all
oF your answers on a separate sheet
oF paper and make sure you have clearly labeled the problem corresponding to your answer. Absolutely
no cheating. Work as quickly and e±ciently as possible so that you can ²nish all oF the problems.
Name:
Some Equations
Poisson
If
X
∼
Pois(
λ
), the pmf of
X
is
p
(
x
)=
λ
x
x
!
e

λ
Hypergeometric
If
X
∼
hypergeom(
N, k, n
), the pmf of
X
is
p
(
x
(
k
x
)(
N

k
n

x
)
(
N
n
)
and E(
X
nk/N
.
Negative Binomial
If
X
∼
NegBin(
r, p
), the pmf of
X
is
p
(
x
±
x
+
r

1
r

1
²
p
r
(1

p
)
x
,x
=0
,
1
,
2
, . . .
Also, E(
X
r
(1

p
)
p
and var(
X
r
(1

p
)
p
2
.
Exponential
If
X
∼
exp(
θ
), the pdf of
X
is
f
(
x
³
1
θ
e

x/θ
≥
0
0
,
else
Gamma
If
X
∼
gamma(
α, β
), the pdf of
X
is
f
(
x
³
1
Γ(
α
)
β
α
x
α

1
e

x/β
≥
0
0
,
else
Also, E(
X
αβ
and var(
X
αβ
2
.
Beta
If
X
∼
beta(
α, β
), the pdf of
X
is
f
(
x
³
Γ(
α
+
β
)
Γ(
α
)Γ(
β
)
x
α

1
(1

x
)
β

1
,
0
<x<
1
0
,
else
Also, E[
X
]=
α
α
+
β
and var(
X
αβ
(
α
+
β
)
2
(
α
+
β
+1)
.
Multinomial
If
X
∼
multinomial(
n
;
p
1
, . . . , p
k
), then
X
has the pmf
P
(
X
1
=
x
1
, . . . , X
k
=
x
k
n
!