Stats 2D03 Test 1 formula sheet
Noah Forman
Discrete Uniform

U
∼
Uniform(
{
1
,
2
, . . . , n
}
)
PMF:
p
U
(
k
) =
1
n
for
k
∈ {
1
, . . . , n
}
,
E
[
U
] =
n
+ 1
2
.
Bernoulli

I
∼
Bernoulli(
p
)
PMF:
p
I
(
k
) =
1
−
p
if
k
= 0
,
p
if
k
= 1
,
for
k
∈ {
0
,
1
}
,
E
[
I
] =
p.
Binomial

S
∼
Binomial(
n, p
)
PMF:
p
S
(
k
) =
n
k
p
k
(1
−
p
)
n
−
k
for
k
∈ {
0
,
1
, . . . , n
}
,
E
[
S
] =
np.
Geometric

G
∼
Geometric(
p
)
PMF:
p
G
(
k
) = (1
−
p
)
k
−
1
p
for
k
∈ {
1
,
2
,
3
, . . .
}
,
E
[
G
] =
1
p
.
Continuous Uniform

U
∼
Uniform[
a, b
]
PDF:
f
U
(
x
) =
1
/
(
b
−
a
)
if
x
∈
[
a, b
]
,
0
otherwise.
CDF:
F
U
(
x
) =
0
if
x < a,
x
−
a
b
−
a
if
x
∈
[
a, b
]
,
1
if
x > b.
E
[
U
] =
a
+
b
2
.
(
x
+
y
)
n
=
n
X
k
=0
n
k
x
k
y
n
−
k
.
P
n
[
j
=1
A
j
!
=
X
J
⊆{
1
,...,n
}
, J
̸
=
∅
(
−
1)

J

+1
P
\
i
∈
J
A
i
!
=
P
(
A
1
) +
P
(
A
2
) +
· · ·
+
P
(
A
n
)
(
n
terms
)
−
P
(
A
1
∩
A
2
)
− · · · −
P
(
A
n
−
1
∩
A
n
)
n
2
terms
+
P
(
A
1
∩
A
2
∩
A
3
) +
· · ·
+
P
(
A
n
−
2
∩
A
n
−
1
∩
A
n
)
n
3
terms
.
.
.
+ (
−
1)
n
+1
P
(
A
1
∩
A
2
∩ · · · ∩
A
n
)
.