Probabilistyka
na podstawie idei teorii niezawodności
Karol Dziedziul
2009
1. Podstawowe pojęcia teorii niezawodności
É " &!
T = T (x)
x
T : &! [0, +").
T
t
F (t) = P {É " &! : T (x)(É) d" t}.
F
0 d" F (t) d" 1 F (t)
t
F (t)
t
F F (0) = 0
R(t) = 1 - F (t) = P {É " &! : T (x)(É) > t}.
F
f
2
f(t) = F (t).
+"
t
F (t) = f(s)ds.
0
f e" 0,
+"
"
f(t)dt = 1,
0
+"
b
P (a < T d" b) = f(s)ds.
a
F
T
f(t)
(t) = -(lnR(t))2 = .
R(t)
t
R(t) - R(t + dt)
(t) H"
dtR(t)
+"
t
R(t) = exp(- (s)ds),
0
+"
t
(s)ds
f(t) = R(t)(t) = e- 0
(t).
+"
"
ET = tf(t)dt.
0
+"
"
k
ET = tkf(t)dt.
0
2
V arT = E(T - ET )2 = ET - (ET )2
"
à = V arT .
EX V arX
2. Podstawowe rozkłady
F (t) = 1 - exp(-t).
 = 3
1
0.8
0.6
0.4
0.2
0.5 1 1.5 2
f(t) = e-t.
  = 3
1
0.8
0.6
0.4
0.2
0.5 1 1.5 2
(t) = 
1 1
ET = V arT = .
 2
t e" 0
(t) = Ä…tÄ…-1,
Ä…,  e" 0 Ä… > 1
Ä… = 1
Ä… < 1
+"
t
R(t) = exp[- (s)ds] = exp[-tÄ…].
0
1 1
ET = “(1 + )- Ä…
Ä…
( )
2 1 2
V arT = “(1 + ) - “2(1 + ) - Ä… .
Ä… Ä…
“
“(n + 1) = n!.
 x > 0
+"
"
“(x) = tx-1e-tdt.
0
“
"
Ä„
“(n + 1/2) = (2n - 1)!!,
2n
!!
5!! = 1 · 3 · 5
 = 4, Ä… = 0.5
1
0.8
0.6
0.4
0.2
0.5 1 1.5 2
Ä…, 
Ä…
f(t) = tÄ…-1exp(-t).
“(Ä…)
 = 4, Ä… = 6
0.025
0.02
0.015
0.01
0.005
1 2 3 4 5 6
 Ä…, 
(t) = Ä…et.
Ä… = 0.0204
 = 0.097
1
0.8
0.6
0.4
0.2
5 10 15 20 25 30 35
1
0.8
0.6
0.4
0.2
5 10 15 20 25 30 35
0.04
0.03
0.02
0.01
5 10 15 20 25 30 35
0.6
0.5
0.4
0.3
0.2
0.1
5 10 15 20 25 30 35
N(0, 1) t " (-", ")
1 t2
f(t) = exp(- ).
2Ä„ 2
Åš N(0, 1)
+"
x
Åš(x) = f(t)dt.
-"
(0, ")
U
EU = 0
V arU = 1
+"
1
P (-1 < U d" 1) = f(t)dt = Åš(1) - Åš(-1) H" 0.68.
-1
0.15
0.125
0.1
0.075
0.05
0.025
-4 -2 2 4
Gęstość rozkładu standardowego
µ, Ã N(µ, Ã)
1 (x - µ)2
f(t) = exp(- ).
2Ä„Ã 2Ã2
X N(µ, Ã)
EX = µ,
"
V arX = Ã.
µ, Ã
N(1, 4)
-2 2
-2 - 1 2 - 1
P (-2 < X d" 2) = P ( < U d" ) = Åš(1/4) - Åš(-3/4).
4 4
 N(µ, Ã)
N(0, 1)
T

ET = 1/

3. Proces stanów na przykładzie procesu defaultu
{
0 0 d" t < T (É)
Xt(É) =
1 T (É) d" t
t Xt 0
1
P (Xt = 1) = F (t)
P (Xt = 0) = R(t).
t Xt
0 1 Xt
T (É) = inf{t e" Xt(É) = 1}.
Xt
{0, 1} T
 T
Xt
F
"
"
"
T
T
F (T )
T
100E(1 - XT ) = 100R(T ).
F (n) n = 1, 2, 3, ,
4. Niezależność i prawdopodobieństwo warunkowe
na przykładzie własności rozkładu wykładniczego
s
P (A )" B)
P (A|B) = .
P (B)
P (T > t + s) R(t + s)
P (T > t + s|T > s) = = .
P (T > s) R(s)

R(t) = exp(-t).
P (T > t + s|T > s) = exp(-t) = R(t)
R(t)R(s) = R(t + s).
s
P (A )" B) = P (A)P (B).
"
"
"
X Y
5. Zastosowanie poznanych narzędzi probabilisty-
cznych w ubezpieczeniach na życie
x = 45
Ax
x = 45 Ax
ax x = 45
Px x = 45
6. System nieodwracalny - proces śmierci
E m
E 1 0
¨
¨ : {0, 1}E {0, 1}.
{0, 1}E
E
¨(1, ..., 1) = 1,
¨(0, ..., 0) = 0,
¨
E
k m k
m
k m w =
(w1, ..., wm) " Rm u = (u1, ..., um) = (w(1), ..., w(m))
w(1) e" ... e" w(m)
w1, ..., wm
k
Mm(w) = uk.
j
j " E Xt
E
S 1 m
¾t = ¨(Xt , ..., Xt ).
k m
S k 1 m
¾t = Mm(Xt , ..., Xt ).
E
S
TS = inf{t " [0, ") : ¾t = 0}.
m m
S
T
S m
T = min{T1, ..., Tm} = Mm (T1, ..., Tm),
Tj j
1 m
S 1
T = max{T1, ..., Tm} = Mm(T1, ..., Tm).
Tj 
max
P (max{T1, ..., Tm} d" t) = P (T1 d" t, ..., Tm d" t)
= P (T1 d" t) · · · P (Tm d" t) = (1 - exp(-t))m.
min
m
P (min{T1, ..., Tm} > t) = P (T1 > t) · · · P (Tm > t)
= (exp(-))m = exp(-m).
k m
S(t)
t Z0, Z1, ...Zm-k+1 Z0
Z1 1
Z0 Z1 Z2 ... Zm-k+1.
Zm-k+1 S(t)
P {S(t) = Zj} = pj(t).
Tj
j Tj,r
Tj,r
r r d" 
r =  S(t) Zj
½j = (m - j).
r = 0 S(t) Zj
½j = k.
 S(t) Zj
½j = k + (m - k - j)r.
pj(t)
t + dt j
t j j - 1 S(t) Zj
½j Zj-1
½j-1
pj(t + dt) H" pj(t)(1 - ½jdt) + pj-1(t)½j-1dt.
|x|
ex H" 1 + x.
Zj t t + dt
Rj(dt) = e-½jdt H" (1 - ½jdt).
Zj-1 Zj t t+dt
Fj-1(dt) = 1 - e-½j-1dt H" -½j-1dt.
pj(t + dt) - pj(t) H" pj(t)½jdt + pj-1(t)½j-1dt
pj(t + dt) - pj(t)
H" pj(t)½j + pj-1(t)½j-1.
dt
p2 (t) = pj(t)½j + pj-1(t)½j-1
j
p0(0) = 1, p1(0) = 0, ..., pm(0) = 0.
k m
m-k
"
RS(t) = pj(t) = 1 - pm-k+1(t).
j=0
 f : [0, ") R
+"
"
F (s) = L(f)(s) = f(t)e-stdt,
0
s
RS
(s + ½0) · · · (s + ½m-k) - ½0 · · · ½m-k
L(RS)(s) = .
s(s + ½0) · · · (s + ½m-k)
7. Proces odnowy
Ä0, Ä1, Ä2 . . .
Ä0 FA
F
t
fA
FA f F
n-1
"
Sn = Äj.
j=0
Fn Sn
+"
t
Fn(t) = Fn-1(t - s)f(s)ds
0
+"
t
fn(t) = fn-1(t - s)f(s)ds.
0
X Y f g
X + Y f " g f g
+"
"
f " g(t) = f(t - s)g(s)ds.
-"
[0, ")
+"
t
f " g(t) = f(t - s)g(s)ds.
0
f g
L(f " g) = L(f)L(g).
fn n =
1, 2...
{Xj}j
EXj = µ V arXj = Ã2
n
"
"
1
Xj H" N(µ, Ã/ n).
n
j=1
Ä0 Äj
"n-1
1
Äj - µ
Sn - µ
j=0
n
" = " H" N(0, 1).
Ã/ n Ã/ n
É t
{
0 t < Ä0(É)
½(t)(É) =
n Sn(É) d" t < Sn+1(É)
P (½(t) d" n - 1) = P (Sn > t) = 1 - P (Sn d" t) = 1 - Fn(t).
P (½(t) = n - 1) = P (½(t) d" n) - P (½(t) d" n - 1) = Fn(t) - Fn+1(t).
½
H(t) = E½(t).
h(t) = H2 (t).
" "
" "
H(t) = E½(t) = jP (½(t) = j) = j(Fj(t) - Fj+1(t))
j=0 j=0
+"
" "
t
" "
Fj(t) = FA(t) + f(s)Fn-1(t - s)ds.
0
j=1 j=2
+"
"
t
"
H(t) = FA(t) + f(s)H(t - s)ds.
0
j=2
H = FA + f " H.
L(H) = L(FA) + L(f)L(H).
L(FA)
L(H) = .
1 - L(f)
8. Prosty proces odnów
H(t)
(2)
h(t) = f(t) + f " h(t).
L(h) = L(f) + L(f)L(h).
L(f)
L(h) = .
1 - L(f)

L(f)(s) = .
 + s


+s
L(h)(s) = = .

s
1 -
+s
h(t) =  oraz H(t) = t.
1

t t