Một số bài toán đặc trưng của phân phối hình học hai chiều

S
F = {F (x, θ), θ ∈ O} X ≈ F ∈ F S
S X F
n
p q = 1 −p)
r r
n r
r ν ν ≤ n
n ν, r, p ν ≥ r
ν = r + k r (r + k), k = 0, 1, 2,
f(k; r, p) k
r (r + k −1)
k (r + k)
C
k
n+k−1
p
r−1
q
k
p
f(k; r, p) = C
k
n+k−1
p
r
q

k
;
k = 0, 1, 2,
r
r
k A r A A
A p
r
q
k
.
r
∞

k=0
f (k; r, p)
r
r
∞

k=0
f (k; r, p) = 1
r
r
r 0 < p < 1 {f(k; r, p)}
r {f(k; r, p)}
r
r = 1
X
X

f(x) = P (X = x) = q
x
p x = 0, 1, 2, 3,
f(x)
f p
EX =
1 − p
p
=
1
p
− 1,
DX =
1 − p
p
2
.
I
+
1
I
+
1
= {0, 1, 2, }
Y
f(y) = P (Y = y) = q
y−1
p y = 1, 2, 3,
I
1

= {1, 2, 3, }.
X = Y − 1
Ge(p)
EY =
1
p
, DY =
1 − p
p
2
.
X 0, 1, 2, P (X = k) = p
k
.
P (s) = p
0
+ p
1
s + p
2
s
2
+ p
3
s
3
+ (−1 ≤ s ≤ 1)
X {p
k
}

P (s) = E(s
X
)
P (X = k) = q
k
p, k = 0, 1, 2,
P (s) = p
∞

k=0
(qs)
k
=
p
1 − qs
.
P (Y = k) = q
k−1
p, k = 1, 2, 3,
P (s) =
ps
1 − qs
=
ps
1 − (1 −p)s
.
ϕ (t) = Ee
itX
, , t ∈ R
1

X.
ϕ(t) = P

e
it

.
ϕ(t) =
p
1 − qe
it
.
Y
f (y) = p
y
(1 − p) ; 0 < p < 1, y = 0, 1, 2,
P [Y ≥ t + s|Y ≥ t] = P [Y ≥ s]
t s
E [Y − t|Y ≥ t] = E (Y )
X
R(t) = P (X > t)
R(t) t
R(t)
r(x) =
f (x)
R(x)
n r
A
1
, , A

r
p
1
, , p
r
0 < p
i
< 1, p
1
+ + p
r
= 1
n A
1
k
1
A
2
k
2
A
r
k
r
(k
1
+ + k
r
= n)
n!

k
1
!k
2
! k
r
!
.p
k
1
1
p
k
r
2
p
k
r
r
.
p
X
p (x
1
, x
2
, , x
p
) = C(N, α)


1 +
p

j=1
θ
j

−α−N
p

j=1
θ
x
j
j
x
j
!
α > 0; θ
j
> 0 (j = 1, 2, , p) ; x
1
, x
2
, , x
p
= 0, 1, 2, ;
C(N, α) N, α θ
j
;

N = x
1
+ x
2
+ + x
p
; N = 0, 1,
p = 2 α = 1
p (x
1
, x
2
) =

x
1
+ x
2
x
1

θ
x
1
1
θ
x
2
2
(1 + θ

1
+ θ
2
)
−(x
1
+x
2
+1)
x
1
, x
2
= 0, 1, 2, ; θ
1
, θ
2
> 0
∅(t) , ψ (t) V
G(v) ∅(t)
∅(t) = ψ (t) E [∅(tv)]
ψ (t)
G(v)
P [V = 0] = a; P [V = 1] = b; a + b = 1; 0 < a ≤ 1.

0 0
0 0

,


0 0
0 0

,

0 0
0 1

,

1 0
0 1

a, b, c, d
a + b + c + d = 1, b + d < 1, c + d < 1 T = (t
1
, t
2
).
∅(t) ψ (t)
ψ (t) = E

e
it
1
X
1
+it
2
X

2

∅(t) = E

e
it
1
Y
1
+it
2
Y
2

.
∅(T ) = ψ (T ) E [∅(T V )] .
ψ (T )
ψ (T ) =

1 +
p
1 − p

1 − e
it
1


−1


1 +
q
1 − q

1 − e
it
2


−1
= ψ
1
(t
1
, o) × ψ
2
(o, t
2
) .
∅(T ) = ψ
1
(t
1
, o) ψ
2
(o, t
2
) [a + b∅(t
1
, o) + c∅(o, t

2
) + d∅(T )] .
t
1
, t
2
= 0
E (Y
i
) = θ
i
V (Y
i
) = θ
i
(1 + θ
i
) ; i = 1, 2
Cov (Y
1
, Y
2
) =
ad − bc
1 − d
θ
1
θ
2
θ

1
= p [(1 − p) (1 − b − d)]
−1
; θ
2
= q [(1 − q) (1 − c − d)]
−1
p (x
1
, x
2
) =

x
1
+ x
2
x
1

p
x
1
1
p
x
2
2
(1 − p
1

− p
2
) .
x
1
, x
2
= 0, 1, 2, ; 0 < p
1
, p
2
< 1; p
1
+ p
2
< 1.
r = 3, k
1
= x
1
, k
2
= x
2
, k
3
= 1
A
3
A

1
x
1
A
2
x
2
A
3
P [X = x + 1|X ≥ 1] = P [X = x]
P [Y ≤ n] − P [X + Y ≤ n] = βP [X + Y = n]
x, n = 0, 1, 2,
P [X
1
= x
1
+ 1, X
2
= x
2
+ 1|X
1
≥ 1, X
2
≥ 1] = P [X
1
= x
1
, X
2

= x
2
]
P [Y ≤ n] − P [X + Y ≤ n] = βP [X + Y = n]
X = (x
1
, x
2
) ; Y = (y
1
, y
2
) ; n = (n
1
, n
2
) , x
1
, x
2
= 0, 1, 2,
P [X = (x
1
, x
2
)] = c
1
P [X = (x
1
− 1, x

2
)] + c
2
P [X = (x
1
, x
2
− 1)]
x
1
, x
2
= 0, 1, 2, (x
1
, x
2
) = (0, 0) ,
c
1
> 0, c
2
> 0, c
1
+ c
2
< 1
P [X = (x
1
, x
2

)] = 0 x
1
< 0, x
2
< 0
P [Y = n] = (1 + β) P [X + Y = n]−θβP [X + Y = n − I
1
]−(1 − θ) βP [X + Y = n − I
2
] .
I
1
, I
2
P (I
i
) = π
i
, i = 1, 2.
p
1
= θ
1
(1 + θ
1
+ θ
2
)
−1
p

2
= θ
2
(1 + θ
1
+ θ
2
)
−1
.
f (x
1
, x
2
) = p
x
1
1
p
x
2
2
θ
x
1
x
2
−1

1 − p

1
θ
x
2
+1

1 − p
2
θ
x
1
+1

+ θ − 1

,
0 ≤ p
1
, p
2
≤ 1, 0 ≤ θ ≤ 1; 1 − θ ≤ (1 − p
1
θ)(1 − p
2
θ)
x
1
, x
2
= 0, 1, 2,

θ = 1
f(x
1
, x
2
) = p
x
1
1
p
x
2
2
(1 − p
1
)(1 − p
2
); x
1
, x
2
= 0, 1, 2,
A
1
x
1
A
1
A
2

x
2
A
2
{X
i
, i = 1, 2, } r = 3
A
1
, A
2
, A
3
P (X
i
= A
1
) = p, P (X
i
= A
2
) = q, P (X
i
= A
3
) = r,
0 ≤ p, q, r p + q + r = 1.
M = min{m : X
m
= A

1
}, N = min{n : X
n
= A
2
}.
(M, N)
f (m, n) = P (M = m, N = n) =



pqr
m−1
(1 − q)
n−m−1
, n > m ≥ 1,
0, m = n,
pqr
n−1
(1 − p)
m−n−1
, m > n ≥ 1.
(M, N)
p q
BGe(p, q)
f(m, n)
A
i
A
1

m A
2
n
M  Ge(p), N  Ge(q)
P
M
(s) =
ps
1 − (1 − p) s
, P
N
(s) =
qs
1 − (1 − q) s
N M
P (N = n|M = m) =







pr
m−1
(1 − q)
n−2m−1
, n > m ≥ 1,
pr
n−1

(1 − p)
m−n−1
(1 − q)
m
m > n ≥ 1, q < p.
R (M, N)
P (R = k) = (1 − r)r
k−1
, k = 1, 2,
(M, N)
G (u, v) =
pquv
1 − ruv

u
1 − (1 − p) u
+
v
1 − (1 − q) v

.
G (u, v) = G
R
(uv)

q
p + q
G
M
(u) +

p
p + q
G
N
(v)

G (u, v) =

1 −
(1 − u) (1 − v)
1 − ruv

G
M
(u) G
N
(v) ,

P
R
(w) R
(M, N)
f (m, n) =










(1 − p) f (m − 1, n) , n ≥ 1, m = n + 2, n + 3,
rf (m − 1, n − 1) ,
m, n = 2, 3,
(1 − q) f (m, n − 1) , m ≥ 1, n = m + 2, m + 3,
f(1, 1) = 0 f(1, 2) = f(2, 1) = pq
min(M, N) R min(M, N)
d
=
R
M +N P
(M,N)
(u, v) u = v
P
(M+N)
(u) = G
R

u
2


q
p + q
G
M
(u) +
p
p + q

G
N
(u)

.
M N
M N
E(MN ) =
−1
1 − r
+
1
pq
cov(M, N) =
−1
1 − r
ρ (M, N) =
−pq
(1 − r)

(1 − p) (1 − q)
.
P (M > m, N > n)
X =
(X
1
, X
2
) I
+

2
= {(x
1
, x
2
) ; x
1
, x
2
= 0, 1, 2, } X
F (x
1
, x
2
) f(x
1
, x
2
)
r (t) = E [X −t|X ≥ t]
r
i
(t
1
, t
2
) = E [X
i
− t
i

|X ≥ t]
t = (t
1
, t
2
) X ≥ t
X
i
≥ t
i
i = 1, 2
r
i
(o, o) = E (X
i
) .
R (t
1
, t
2
) = P [X
1
≥ t
1
, X
2
≥ t
2
] .
R(t

1
, t
2
)
r
i
(t
1
, t
2
) R (t
1
, t
2
) =
∞

t
1
∞

t
2
(x
i
− t
i
) f (x
1
, x

2
)
=
∞

t
1
(x
i
− t
i
) P [X
1
= x
1
, X
2
≥ t
2
]
=
∞

s=1
sP [X
1
= t
1
+ s, X
2

≥ t
2
]
=
∞

s=1
R (t
1
+ s, t
2
).
t
1
(t
1
+ 1)
R (t
1
, t
2
) r
1
(t
1
, t
2
) − R (t
1
+ 1, t

2
) [1 + r
1
(t
1
+ 1, t
2
)] = 0.
r
2
(t
1
, t
2
)
R (t
1
, t
2
) r
2
(t
1
, t
2
) − R (t
1
, t
2
+ 1) [1 + r

2
(t
1
, t
2
+ 1)] = 0.
r (t) = (r
1
(o, o) , r
2
(o, o)) = (r
1
, r
2
)
r
1
r
2
t
1
t
2
t
1
, t
2
≥ 0.
E [X
i

− t
i
|X ≥ t] = E [X
i
|X
j
≥ t
j
]
; i, j = 1, 2; i = j.
r
i
(t
1
, t
2
) = a
i
(t
j
) ;
i, j = 1, 2, i = j
a
i
(t
j
) t
i
f
i

(x
i
) = (1 − p
i
) p
x
i
i
;
x
i
= 0, 1, 2, ; i = 1, 2; 0 < p
i
< 1.
f
i
(x
i
)
E (X
1
HmphnphicØiukinca
i
X
j
= t
j
f (X
i
|X

j
= t
j
) = (1 − p
j
)
−1
p
x
i
i
θ
x
i
t
j
−1

1 − p
j
θ
x
i
+1

1 − p
i
θ
t
j

+1

+ θ − 1

2
= t
2
= α
2
r
1
(t
1
, t
2
) + (1 − α
2
) r
1
(t
1
, t
2
+ 1)vE (X
2
|X
1
= t
1
) = α

1
r
2
(t
1
, t
2
) +
(1 − α
1
) r
2
(t
1
+ 1, t
2
) α
j
= 1 + E (X
j
) r
j
(t
1
, t
2
) ;
j = 1, 2
E (X
1

, X
2
) =
∞

x
2
=1
p
1
(p
2
θ)
x
2
1 − p
1
θ
x
2
.
X
1
X
2
r
X
1
,X
2

=
A (θ; p
1
, p
2
) −
p
1
1 − p
1
p
2
1 − p
2

p
1
(1 − p
1
)
2
p
2
(1 − p
2
)
2

1/2
= [(1 − p

1
) (1 − p
2
) A (θ; p
1
, p
2
) − p
1
p
2
] (p
1
p
2
)
−
1
2
,
A (θ; p
1
, p
2
)
p
1
p
2
r θ

dr
dθ
.
0 −
√
p
1
p
2
0 θ = 1
X
1
X
2
X
1
X
2
θ = 1
X
i
X
j
≥ t
j
P [X
i
≥ x
i
|X

j
≥ x
j
] =

p
i
θ
t
j

x
i
, i = 1, 2
f (X
i
|X
j
≥ t
j
) =

p
i
θ
t
j

x
i


1 − p
i
θ
t
j

x
i
= 0, 1, 2,

p
i
θ
t
j

.
(r, s)
µ

rs
=

x
1
x
r
1
[A(x

1
, s) − A(x
1
+ 1, s)],
A (s
1
, s) = p
x
1
1
(1 − p
2
θ
x
1
)

x
2
x
s
2
(p
2
θ
x
1
)
x
2

.
E

s
X
1
1
s
X
2
2

=

x
1
s
x
1
1
[B (x
1
, s
2
) − B (x
1
+ 1, s
2
)],
B (x

1
, s
2
) = p
x
1
1
(1 − p
2
θ
x
1
) (1 − s
2
p
2
θ
x
1
)
−1
.
r X
i
φ
(r)
i
(t
1
, t

2
) = E

(X
i
− t
i
)
(r)
|X
1
≥ t
1
, X
2
≥ t
2

, i = 1, 2,
(X
i
− t
i
)
(r)
= (X
i
− t
i
) (X

i
− t
i
− 1) (X
i
− t
i
− r + 1) .
p
i
= 1−α
i
n
i
; i = 1, 2; θ= 1−α
n
n = n
1
n
2
R

x
1
n
1
,
x
2
n

2

n
1
n
2
0
R(x
1
, x
2
) = p
x
1
1
p
x
2
2
θ
x
1
.x
2
R

x
1
n
1

,
x
2
n
2

= (1 − α
1
n
1
)
x
1
/n
1
(1 − α
2
n
2
)
x
2
/n
2
(1 − α
n
)
x
1
x

2
/n
.
n
1
, n
2
R (x
1
, x
2
) = exp [−α
1
x
1
− α
2
x
2
− αx
1
x
2
]
x
1
, x
2
> 0; α
1

, α
2
> 0; α ≥ 0.
f (x
1
, x
2
) = [(α
2
+ θx
1
) (α
1
+ θx
2
) − θ] exp (−α
1
x
1
−α
2
x
2
− θx
1
x
2
)
x
1

, x
2
> 0 α
1
, α
2
> 0; θ > 0
I
+
2
X
1
X
2
R (t
1
+ 1, t
2
) =
r
1
1 + r
1
R (t
1
, t
2
) .
t
1

R (t
1
, t
2
) =

r
1
1 + r
1

t
1
R (o, t
2
)
t
2
= 0
R (t
1
, 0) =

r
1
1 + r
1

t
1

.
R (0, t
2
) =

r
2
1 + r
2

t
2
.
R (t
1
, t
2
) =

r
1
1 + r
1

t
1

r
2
1 + r

2

t
2
r
i
= E (X
i
) ,
R (t
1
, t
2
) = R (t
1
, o) R (o, t
2
)
P [X
1
≥ t
1
, X
2
≥ t
2
] = P [X
1
≥ t
1

] P [X
2
≥ t
2
]
(t
1
, t
2
) I
+
2
X
1
X
2
X
i
r
i
X
1
X
2
p
1
, p
2
r
i

= p
i
(1 − p
i
)
−1
, i = 1, 2 t
1
t
2
t
1
, t
2
X = (X
1
, X
2
) I
+
2
a
1
(t
2
) a
2
(t
1
)

a
i
(o) = p
i
(1 − p
i
)
−1
, i = 1, 2.
X
R (x
1
, x
2
) = p
x
1
1
p
x
2
2
θ
x
1
x
2
; x
1
, x

2
= 0, 1, 2
0 ≤ p
1
, p
2
≤ 1; 0 ≤ θ ≤ 1; 1 − θ ≤ (1 − p
1
θ) (1 − p
2
θ)
X X
F (x
1
, x
2
) = 1 − p
x
1
+1
1
− p
x
2
+1
2
+ p
x
1
+1

1
p
x
2
+1
2
θ
(x
1
+1)(x
2
+1)
f (x
1
, x
2
) = p
x
1
1
p
x
2
2
θ
x
1
x
2
−1


1 − p
1
θ
x
2
+1

1 − p
2
θ
x
1
+1

+ θ − 1

.
i = 1 f(x
1
, x
2
)
r (t
1
, t
2
) R (t
1
, t

2
) =
∞

t
1
∞

t
2
(x
1
− t
1
)f (x
1
, x
2
)
=
∞

t
1
(x
1
− t
1
) p
x

1
1

(1 − p
2
θ
x
1
)
∞

t
2
(p
2
θ
x
1
)
x
2
− p
1

1 − p
2
θ
x
1
+1


∞

t
2

p
2
θ
x
1
+1

x
2
