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Probability Mass Function for Discrete Variables
•
Binomial distribution,
Bin
(
x
;
n, p
)
:
P
(
X
=
x
) =
n
x
p
x
(1
−
p
)
n
−
x
where
x
= 0
,
1
, . . . , n
•
Poisson distribution,
Po
(
x
;
λ
)
:
P
(
X
=
x
) =
e
−
λ
λ
x
x
!
where
x
= 0
,
1
,
2
, . . .
•
Hypergeometric distribution,
H
(
x
;
N, n, k
)
:
P
(
X
=
x
) =
N
−
k
n
−
x
k
x
N
n
where
x
= 0
,
1
,
2
, . . . , n
and
max(0
, k
+
n
−
N
)
≤
x
≤
min(
n, k
)
•
Geometric distribution,
Geom
(
x
;
p
)
:
P
(
X
=
x
) =
p
(1
−
p
)
x
−
1
where
x
= 1
,
2
,
3
, . . .
•
Negative Binomial distribution,
Bin
∗
(
x
;
r, p
)
:
P
(
X
=
x
) =
x
−
1
r
−
1
p
r
(1
−
p
)
x
−
r
where
x
=
r, r
+ 1
, r
+ 2
, . . .
Probability Density Function for Continuous Variables
•
Normal distribution,
N
(
x
;
µ, σ
2
)
:
f
(
x
) =
1
σ
√
2
π
exp
−
(
x
−
µ
)
2
2
σ
2
where
− ∞
< x <
∞
•
t
distribution,
t
(
x
; Γ
, ν
)
:
f
(
x
) =
Γ
ν
+1
2
√
πν
Γ
ν
2
1 +
x
2
ν
−
ν
+1
2
where
− ∞
< x <
∞
•
Chi-Squared distribution,
χ
2
(
x
; Γ
, k
)
:
f
(
x
) =
x
(
k/
2)
−
1
exp
−
x
2
2
(
k/
2)
Γ
k
2
where
x
≥
0
•
F
distribution,
f
(
x
;
ν
1
, ν
2
)
:
f
(
x
) =
1
β
ν
1
2
,
ν
2
2
ν
1
ν
2
ν
1
2
x
(
ν
1
/
2)
−
1
1 +
ν
1
ν
2
x
−
ν
1 +
ν
2
2
where
x
≥
0
3
Statistical Tables and Formulae 2.0