• X ∼ Ber(p)
• f (x) = p x(1 − p)1−x, x = 0, 1
• E(X ) = p
• Var(X ) = p(1 − p)
• X ∼ B(p)
• f (x) =
n
p x(1 − p)n−x, x = 0, 1, . . . , n
(x)
• E(X ) = np
• Var(X ) = np(1 − p)
• X ∼ Ge(p)
• f (x) = p(1 − p)x−1, x = 1, 2, . . .
1
E(X
)
=
•
p
1−p
Var(X
)
=
•
p2
• 有無記憶性質：P(X > s + t | X > s) = P(X > t) s, t = 1, 2, . . .
• X ∼ NB(r, p)
x−1
p r (1 − p)x−r, x = r, r + 1, . . .
(r − 1)
r(1 − p)
E(X
)
=
•
P
r(1 − p)
Var(X
)
=
•
P2
• f (x) =
• X ∼ H(N, D, n)
( x )( n − x )
D
•
f (x) =
N−D
N
(n)
, ma x{0, n − N + D} ≤ x ≤ min{n, D}
nD
N
nD(N − n)(N − D)
Var(X
)
=
•
N 2(N − 1)
• E(X ) =
• X ∼ P(λ)
e −λ λ x
• f (x) = x! , x = 0, 1, . . .
• E(X ) = λ
• Var(X ) = λ
•X ∼ D − U(N1, N2)
1
, x = N1, N1 + 1, . . . , N2
N2 − N1 + 1
N+1
E(X
)
=
•
2
N2 − 1
• Var(X ) = 12
• f (x) =
•X ∼ U[a, b]
1
, x ∈ [a, b]
b−a
b+a
• E(X ) = 2
(b − a)2
• Var(X ) =
12
• f (x) =
•X ∼ Γ[α, β]
−x
x α−1e β
• f (x) = Γ(α)β α , x > 0
∞
α−1 −t
• Γ(α) = ∫ t e dt, α > 0
•∫
∞
0
0
−x
x α−1e β d x = Γ(α)β α, α, β > 0
• Γ(α + 1) = αΓ(α), α > 0
• Γ(1) = 1
• Γ(n) = (n − 1)!, n ∈ ℤ+
• E(X ) = αβ
• Var(X ) = αβ 2
p
2
•X ∼ χp = Γ( , 2)
•
f (x) =
1
p
Γ( 2 )2
2
p
p
2
−x
x 2 −1e 2 , x > 0
• E(X ) = p
• Var(X ) = 2p
1
•X ∼ ε(λ) = Γ(1, ) = W(λ, 1)
λ
• f (x) = λe −λx, x > 0
1
E(X
)
=
•
λ
1
Var(X
)
=
•
λ2
e −λ(s+t)
• 有無記憶性質：P(X > s + t | X > s) = e −λs = P(X > t), s, t > 0
•X ∼ D − ε(λ)
λ −λ|x|
,x∈ℝ
• f (x) = 2 e
• E(X ) = 0
• Var(X ) =
2
λ2
•X ∼ W(α, β)
β
• f (x) = αβx β−1e −αx , x > 0
−1
1
β Γ(1 +
E(X
)
=
α
)
•
β
−2
2
1 2
β [Γ(1 +
Var(X
)
=
α
)
−
(Γ(1
+
)) ]
•
β
β
•X ∼ N(μ, σ 2)
•
f (x) =
1
2πσ
e
−(x − μ) 2
2σ 2
,x∈ℝ
• E(X ) = μ
• Var(X ) = σ 2
•X ∼ N(0, 1)
•
f (x) =
1
2π
• E(X ) = 0
• Var(X ) = 1
x−μ
•z = σ
−x 2
e 2 ,x∈ℝ
•log X ∼ N(μ, σ 2)
•
1
f (x) =
2πσx
• E(X ) = e
e
−(log x − μ) 2
2σ 2
,x∈ℝ
2
μ+ σ2
2
• Var(X ) = e 2(μ+σ ) − e 2μ+σ
2
•Be(α, β)
Γ(α + β ) α−1
x (1 − x) β−1, 0 < x < 1
Γ(α)Γ(β )
α
E(X
)
=
•
α+β
αβ
Var(X
)
=
•
(α + β )2(α + β + 1)
• f (x) =
•X ∼ C(θ, a)
• f (x) =
a
, x ∈ ℝ, a > 0, θ ∈ ℝ
π(a 2 + (x − θ)2)
•Generating function : g(s) = E(s X ), | s | ≤ 1, r . v . ∈ ℤ+ ∪ {0}
•Laplace transform : ϕ(s) = E(e −sX ), s ≥ 0, r . v . ≥ 0
k
d
k
•E(X ) = k M(t) |t=0
dt