estadistica actuarios

7/30/2019 estadistica actuarios

1/75

Estadstica actuarial: modelos estocsticos

Eva Ferreira

Mara Araceli Garn

03-10

ISBN:


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P( ) = 0,3P( ) = 0,7


8/75

J 30 %70 % S

J %S %

JS

J

J S

P(A2/A1)P(A1)

%

% %

%

XX {0, 1, ..., 10}

X


9/75

XY

X Y < 1200 1200

F(y) = 1 e0,16y, y 0Y

Y N(m = 6,25, 2 = 39,06)

Y

Y U(a, b)

Y


10/75

X p = 0,005n = 2000

X P(X = 0)P(X = 1) P(X = 2) P(X = 3) P(X = 4) P(X = 5)

X [5, 15]

X P( = 10)

X P(X = 0)P(X = 1) P(X = 2) P(X = 3) P(X = 4) P(X = 5)

X [5, 15]

50 %13

810

210


11/75

f(y) =k

(1 + y)3, y 0

f(y) = 0 y < 0

k

n n1n2

X : Y|XX X

Y|X

Y


12/75


13/75

X X


14/75

P()

P(X = k) =ek

k!, k Z+

b(p, n) np

n = np

F(x) = 0, x < 0 F(x) =e

[x]k=0

k

k!x 0

F(0) = P(X 0) = e

X(u) = e(eiu1)

X(u) = e(eu1)

X(u) = (eu 1)

E(X) = V(X) =

sl

X P()

Y X|X > 0 Y{1, 2, 3, ...,}

P(Y = k) = P(X = k|X > 0) =P(X = k

X > 0)

P(X > 0) =

ek

k!(1 e) ,


15/75

k

1 Y

E(Y) =

(1 e)V(Y) =

(1 e) 2e

(1 e)2

Y(u) =e

1 e [eeu 1]

Y X|X l Y{0, 2, 3,...,l 1, l}

P(Y = k) = P(X = k|X l) = P(X = k X l)P(X l) =

k

k!l

j=0j

j!

,

0 k l

Y X|X s Y {s, s +1, s + 2, s + 3,...

}P(Y = k) = P(X = k|X s) = P(X = k X s)

P(X s) =k

k!

j=sj

j!

,

s k

Y X|s X l Y{s, s + 1, s + 2,...,l 1, l}

P(Y = k) = P(X = k|s X l) = P(X = k s X l)P(s X l) =

=k

k!l

j=sj

j!

,

s k l

Y

X


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0 1

P(Y = 0) = P(X = 0) + P(X = 1)

P(Y = 1) = P(X = 1) P(X = 1)P(Y = k) = P(X = k), k 2

E(Y) = E(X) P(X = 1) = (1 e)E(Y2) = E(X2) P(X = 1) = + 2 e

V(Y) = E(Y2)

(E(Y))2 =

e(1 + e

2)

ZY Y = X|X r

Y

P(Y = k) =k

k!j=r

j

j!

,

k r YZ = Y r

P(Z = k) = P(Y = r + k) =r+k

(r + k)!

j=rj

j!

,

k 0

A Ac

Xn = 1 p = P(Xn = 1)

Xn = 0 q = 1 p = P(Xn = 0)

{Xn}nN


17/75

n Yn Y = X1+...+Xn

n pn p

Y = np

p

mm = 1

Ymm m p

Ym Z+

{Ym = k}k m

{Ym = k} = {Sm+k1 = m 1, Xm+k = 1}

m + k m + k 1m 1 k

Sm+k1 = X1 + X2 + ... + Xm+k1 m + k 1{Xn}

m + k 1 p Xm+k

P(Ym = k) = P(Sm+k1 = m 1)P(Xm+k = 1) ==

m + k 1

m 1

pm1qkp =

=

m + k 1

k

pmqk =

= m

k

pm(q)k

m + k 1

k

=

m + k 1

m 1

=

mk

(1)k

k Z+


18/75

Xm p X bn(m, p)

P(X = k) =

m + k 1

k

pmqk =

mk

pm(q)k, k Z+

m

1 =

k=0

P(X = k) = pm

k=0

mk

(q)k =

= pm

k=0

m + k 1

k

qk = pm

k=0

m + k 1

m 1

qk

k=0

mk

(q)k =

k=0

m + k 1

k

qk = pm = (1 q)m

X(u) = E(eiuX) =

k=0

eiukP(X = k) =

=

k=0

eiuk

m + k 1k

pmqk =

= pm

k=0

m + k 1k (eiuq)k == pm(1 eiuq)m

X(u) = pm(1 euq)m

X(u) = mlnp mln(1 euq)

E(X) =mq

p

V(X) =mq

p2


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mp

m

m

m + k 1

k

=

(m + k 1)(m + k 2) (m + k k)k!

m + k 1k

=

m + k 1m 1

= 1

m + k 1 < k 0 < m < 1 k = 0 0! = 10 < x < 1 x! = 1

X X bn(m, p)

pk = P(X = k) =(m + k 1)q

kP(X = k 1) = (m + k 1)q

kpk1


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X

N a ba + b = N

c

c = 0

c = 1

X n

c = 0 X n p = aN

X b(n, a

n)

c = 0 Xc = 0 bli i

ngi i

r

P(bl1) =a

N

P(bl2|bl1) = a + cN + c

P(bl3|bl1 bl2) = a + 2cN + 2c

... = ...

P(blr|bl1 bl2 ... blr1) = a + (r 1)cN + (r 1)c

P(bl1 bl2 ... blr1 blr) == P(bl1)P(bl2|bl1)P(bl3|bl1 bl2) P(blr|bl1 bl2 ... blr1) ==

a

N

(a + c)

(c + N) (a + (r 1)c)

(N + (r 1)c)

s s


21/75

P(ngr+1|bl1 bl2 ... blr) = bN + rc

P(ngr+2|bl1 bl2 ... blr ngr+1) = (b + c)N + (r + 1)c

... = ...

P(ngn|bl1 bl2 ... blr ngr+1... ngr+(s1)) =b + (s 1)c

N + (n 1)c

n r s

r + s = n

P(bl1 bl2 ... blr1 blr ngr+1... ngr+s) == P(bl1)P(bl2|bl1) P(blr|bl1 bl2 ... blr1)P(ngr+1|bl1 bl2 ... blr)...... P (ngr+s|bl1 bl2 ... blr ngr+1... ngr+(s1)) ==

a

N

(a + c)

(N + c) (a + (r 1)c)

(N + (r 1)c)b

(N + rc)

(b + c)

(N + (r + 1)c) (b + (s 1)c)

(N + (n 1)c)

n r

s n

r

= n

s

nr s = n r

Xr n = r + s

P(X = r) =

nr

a

N

(a + c)

(N + c) (a + (r 1)c)

(N + (r 1)c)b

(N + rc)

(b + c)

(N + (r + 1)c) (b + (s 1)c)

(N + (n 1)c)

c = 0

P(X = r) =

nr

acNc

(ac

+ 1)

(Nc

+ 1) (

ac

+ (r 1))(N

c+ (r 1))

bc

( Nc

+ r)

( bc

+ 1)

(Nc

+ (r + 1)) (

bc

+ (s 1))(N

c+ (n 1))

P(X = r) =

ac

+ r 1r

bc

+ s 1s

Nc

+ n 1n


22/75

Xp q

{0, 1, 2..., n}

P(X = k) =

p

+ k 1k

q

+ s 1s

1

+ n 1n

p q = 1 p s = n k

p = aN

q = bN

= cN

X p

pk = P(X = k) =(p

+ k 1)

k

s + 1

( q

+ s)P(X = k 1)

s = n k

p0 = P(X = 0) =

0@q

+ n

1n

1A0@

1

+ n 1n

1A

E(X) = np

V(X) = npq1 + n

1 +


23/75

Xx = 0,3 s2 = 1,164 n = 10

XP(X = 0) P(X = 1) P(X = 2) P(X = 3)

XP(X = 0) P(X = 1) P(X = 2) P(X = 3)

XP(X = 0) P(X = 1) P(X = 2) P(X = 3)

XP(X = 0) P(X = 1) P(X = 2) P(X = 3)

X

X

X n = 10 n = 100

n = 1000

Y = X|X > 0X P()

A B


24/75

X

p = 0,1 Y

X p = 0,1m = 3 X Y

P(Y = 3)

E(Y)

V(Y)

X p m

P(X = k) = pk = pk1(1 p)(m + k 1)/k

XEX = 0,04 V(X) = 0,0504

X P(X = 0) P(X = 1) P(X = 2)P(X = 3) P(X

4)

X = 0,04

X P(X = 0) P(X = 1)P(X = 2) P(X = 3) P(X 4)

X

X1 X21 2 X1 + X2

Z = X1+...+Xn X1,...,Xn1,...,n

Xp = 0,01 = 0,1

n = 4X

P(X = 0) P(X = 1) P(X = 2) P(X = 3) P(X 4)X

xi

ni


25/75

x = 0,04s2 = 0,0792

X XP(X = 0) P(X = 1) P(X = 2) P(X = 3) P(X 4)

X XP(X = 0) P(X = 1) P(X = 2) P(X = 3) P(X 4)

X X

P(X = 0) P(X = 1) P(X = 2) P(X = 3) P(X 4)X

Xx = 0,3 s2 = 1,164

XP(X = 0) P(X = 1) P(X = 2) P(X = 3)

X

P(X = 0) P(X = 1) P(X = 2) P(X = 3)X

P(X = 0) P(X = 1) P(X = 2) P(X = 3)

XP(X = 0) P(X = 1) P(X = 2)

P(X = 3)


26/75


27/75

XY = lnX N(m, 2)

X

T N(0, 1)

f(t) =12

et2

2

t R


28/75

Y

N(m, 2) T Y = T+ m

fY(y) = ft(t(y))|t(y)|ft T t(y) =

ym

t(y) = 1

> 0Y

f(y) =12

e(ym)2

22

y RY = lnX X = eY

X

f(x) = fY(y(x)) |y(x)| ==

12

e(y(x)m)2

22 |y(x)|

y(x) = lnx |y(x)| = | 1x|X = eY

x R+

XR+

f(x) =1

2xe(lnxm)2

22 , x 0

F(x)N(0, 1)

F(x) = P(X x) = P(eY x) = P(Y lnx) = ( lnx m

)

E(X) =

0

xf(x)dx =

0

x1

2xe(lnxm)2

22 dx

t = lnxm

x = et+m dx = et+mdt

E(X) =1

2

et2

2 et+mdt = em1

2

e(t22t)

2 dt


29/75

(t22t+22)

2

= (t)2

2 2

2

E(X) = em+2

212

eh2

2 dh = em+2

2

h = t dh = dtE(X2)

X

V(X) = e2m+2

(e2 1)

Xx

P(X > x) = (k

x)

x k P(X > x) = 1 k

X

F(x) = P(X x) = 1 P(X > x) = 1 ( kx

)

x k F(x) = 0

X k, [k, )

f(x) =k

x+1

k

E(X) =k

( 1)

> 1


30/75

V(X) =k2

( 2)( 1)2

> 2

{x1,...,xn}E(X) x

k

( 1) = x

M =x

(x k)

{x1,...,xn}

L(x, ) = (

k)n

ni=1

(k

xi)+1

lnL(x, ) = n(ln lnk) +n

i=1

( + 1)(lnk lnxi)

MV =1

(lnx lnk)

lnx = 1n

ni=1 lnxi


31/75

Yk, X = Y k

F(x) = P(X x) = P(Y k x) = P(Y x + k) = 1 ( kx + k

)

x

0 F(x) = 0

X

F(x) = P(X x) = 1 ( kx + k

)ebx

x 0 F(x) = 0 , k b

X Z Z = X > 0

Z = X1

F(z) = P(Z z) = P(X1 z) = P(X z) = 1 ( kz + k

)

z 0 F(z) = 0

Z k, r Z

E(Zr) = kr

( r/)(r/+ 1)()

r <


32/75

X a > 0 r > 0(a, r)

f(x) =ar

(r)xr1eax, x 0

f(x) = 0 (r) =0 x

r1exdx

r Z+ (r) = (r 1)!

E(X) =r

a

V(X) =r

a2

X(u) = (1 ua

)r

X(u) = (1 iua

)r

X(u) = r ln(1 ua

)

r = 1


33/75

X a > 0(a)

f(x) = aeax, x 0

f(x) = 0

E(X) = 1a

V(X) =1

a2

X(u) = (1 u

a )1

X(u) = (1 iua

)1

X(u) = ln(1 ua

)

a a r


34/75

X a X E(a)X Y = X + k k > 0 Y

fY(y) = fX(y k) = aea(yk)

y k fY(y) = 0

Xa > 0 k > 0 E(a, k)

f(x) = aea(xk), x k > 0f(x) = 0

E(X) =1

a+ k

a

V(X) =1

a2

X(u) = eiuk(1 iu

a)1

> 0 1a

> 0


35/75

X > 0 > 0 W(, )

f(x) =

x1e(

x) , x 0

f(x) = 0

(r) =0 x

r1exdx r > 0

E(X) =

0

xx1e(x )dx = ( 1

+ 1)

V(X) = 2[(2

+ 1) 2( 1

+ 1)]

(a, ri) i = 1, 2 Y1 =X1

X1+X2

Y2 =X2

X1+X2

Y1 r1, r2Y2 r2, r1

X1 X2 aX = X1 + X2 (a, r1 + r2)

a

Z1 = X1/X

Z R2 R2

(X1, X) (Z1, Z2)Z1 = X1/X Z2 = X

Z =

1X

00 1

X = A X,

||A|| = 1|X| = 0

X = A1 Z = X 00 1

Z,


36/75

X1 = Z1X1 = Z1Z2 X2 = Z2

XZ

||D(X)

D( Z)|| = || Z2 Z1

0 1|| = |Z2|

|x(z)|Z = (Z1, Z2) (X1, X)

fZ(z1, z2) = fX(x1(z1, z2), x(z1, z2))||D( X)

D( Z)|| =

= fX(z1z2, z2)|z2|

Z1 = X1/X

f1(z1) =

f(z1, z2)dz2 =

=

fX(z1z2, z2)|z2|dz2 =

=

0fX(z1z2, z2)z2dz2

z2 = x2 0

(X1, X)

F(x1, x) = P(X1 x1, X x) = P(X1 x1, X1 + X2 x) == P(X1 x1, X2 x x1) = F1(x1)F2(x x1)

X1 X2

fX

(x1, x) = f1(x1)f2(x

x1)

f1 X1 (a, r1) f2 X2 (a, r2)

fX(x1, x) =ar1

(r1)xr111 e

ax1 ar2

(r2)(x x1)r21ea(xx1) =

=ar1ar2

(r1)(r2)xr111 (x x1)r21eax

fX(z1z2, z2) =

ar1ar2

(r1)(r2) (z1z2)

r11

(z2 z1z2)r21

e

az2


37/75

Z1

f1(z1) =

f(z1, z2)dz2 =

=

0

fX(z1z2, z2)z2dz2 =

=

0

ar1ar2

(r1)(r2)(z1z2)

r11(z2 z1z2)r21eaz2z2dz2 =

=ar1+r2

(r1)(r2)zr111 (1 z1)r21

0

eaz2zr1+r212 dz2 =

=ar1+r2

(r1)(r2)zr11

1

(1

z1)r21 (r1 + r2)

ar1+r2=

=(r1 + r2)

(r1)(r2)zr111 (1 z1)r21

0 z1 1 f1(z1) = 0

X r1 > 0 r2 > 0B(r1, r2)

f(x) =

(r1 + r2)

(r1)(r2) xr11

(1 x)r21

=

=1

B(r1, r2)xr11(1 x)r21

0 x 1 f(x) = 0

B(r1, r2) =

10

xr11(1 x)r21dx = (r1)(r2)(r1 + r2)

(r) =0 xr1exdx r > 0

E(X) =r1

r1 + r2

V(X) =r1r2

(r1 + r2)2

(r1 + r2 + 1)


38/75

k

c > kc

X Y = X|X > cY k = c

Xk = 3 = 2,6

6000

X

6000

60009000

X F(x, )

P() =0 [P(X > x)]

1

dx =0 [1 F(x, )]

1

dx

X exp() P() =

X Pareto(, ) P() = () >

M X


39/75

X

M

X > M X MY = mn{X, M} Z = max{0, X M}

FY(y) Y = mn{X, M}

FY(y) = { F(x), x M1, x M

E(Yn) =0 [mn{X, M}]nf(x)dx

Y = mn{

X, M}

={

x, 0 x MM, x M

E([mn{X, M}]n) = M0 xnf(x)dx + Mn[1 F(M)]E(Y) =

M0 xf(x)dx + M[1 F(M)]

X (, ) E(Yn) E(Y)

X > k k = 2E(X) = 3

X

X

f(2), f(3), f(4) f(10)

X (a, r) c

cX (a

c , r)

na

n

n


40/75

T p

1,7 TB V B V T

T

X B V B V T

P(X < 28) = 0,3

E(X|X > 28) =

28xf(x)dx = 21,6

BT

E(B)

T


41/75

k

k X1 + ... + Xk

X YZ+

P(X = j) = aj

P(Y = k) = bk

j, k = 0, 1, 2,...(X = j, Y = k)

P(X = j, Y = k) = P(X = j)P(Y = k) = ajbk


42/75

Z = X + Yr Z+

{Z = r} = {X+ Y = r} = rj=0{X = j, Y = r j}

X Y

P(Z = r) = P(X+ Y = r) =r

j=0

P(X = j, Y = r j) =

=r

j=0

ajbrj = cr

r Z+{cr} =

rj=0 ajbrj {ar} {br}

{cr} = {ar} {br} ZX Y

X Y

P(X = k) =ek

k!, k Z+

Z = X + Y

P(Z = r) = P(X + Y = r) =r

j=0

P(X = j)P(Y = r j) =

=r

j=0

ej

j!

erj

(r j)! =e2r

r!

rj=0

r!

j!(r j)! =

=e2r

r!

rj=0

r

j

=

e2r

r!(1 + 1)r =

e2(2)r

r!

r = 0, 1, 2,... Z 2

F(x) G(y) Z = X + YZ

H(z) = P(Z z) = P(X+ Y z)

{X+Y z} (X, Y)(X, Y)

P(X+ Y

z) = x+yz

f(x)g(y)dxdy =

zx

f(x)g(y)dydx


43/75

H(z) = P(X + Y z) =

zx

f(x)g(y)dydx =

=

f(x)(

zx

g(y)dy)dx =

f(x)G(z x)dx

X Y

H(z) = P(X+ Y z) =

zy

f(x)g(y)dxdy =

=

g(y)(zy

f(x)dx)dy =

g(y)F(z

y)dy

H(z) =

f(x)G(z x)dx =

=

g(y)F(z y)dy

Z

h(z) = H(z) =

f(x)G(z x)

zdx

=

g(y)F(z y)

zdy

h(z) =

f(x)g(z x)dx =

g(y)f(z y)dy

X Y a1a

f(x) = aeax, x R+Z = X+ Y

h(z) =

f(x)g(z x)dx = a2z0

eaxea(zx)dx =

= a2eazz0

dx = a2zeaz, z R+

(a, 2)


45/75

(a + 1)k+r

(k + r)

0

k+r1e(a+1)d = 1

(a + 1, k + r) (k + r) =(k + r 1)(k + r 1) k + r Z+

P(x) =ar

(r)k!

(k + r)

(a + 1)k+r=

r + k 1k

a

a + 1r

1

a + 1k

m = rp = a

(a+1)

X (N, p) X B(N, p)

P(X = k; N) =

Nk

pk(1 p)Nk, k {0,...,N}

NN

P(X = k; N) = P(X = k|N = n) = P(X = k, N = n)P(N = n)

XN N

P(N = n) =

en

n! , n = 0, 1, 2,...


46/75

X N

P(X = k) =

n

P(k|N = n)P(N = n) = (k n)

=

n=k

nk

pk(1 p)nk e

n

n!=

= ek

n=k

n!

k!(n k)!pk(1 p)nknk

n!=

=e(p)k

k!

n=k(1 p)nknk

(n k)! =

=e(p)k

k!

n=0

(q)n

(n)!=

e(p)k

k!eq =

ep(p)k

k!, k = 0, 1,...

p p

X

P(X = k; ) = e

k

k!

f()

X

P(X = k) =

0

P(X = k|)f()d =0

ek

k!f()d

X

SF(x) s F(s) = P(S s)

s

X

s


47/75

S F()(s)

= max S

(s) = P(max S s) =

j=0

P(max S s|X = j) P(X = j)

P(max S s|X = j) = P(max{S1,...,Sj} s) = P(S1 s ... Sj s) = [P(S s)]j

(s) = P(X = 0) + P(X = 1)P(S s) + P(X = 2)P(S s)2 + ... =

=

j=0

P(X = j)P(S s)j =

j=0

P(X = j)F(s)j

X

GX(u) GX(u) = E(uX) X

G(u) =

j=0

P(X = j)uj

(s) = G(F(s))

X S X

X

(s) =

j=0

P(X = j)(F(s))j =

=

j=0

ej

j!F(s)j =

j=0

e(F(s))j

j!=

= eeF(s) = e(1F(s))

F(s) s


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a

Zk = Y1 + ... + Ykk Yi

Zk (a, k) E(Zk) =ka

V(Zk) =ka2

XZ

E(Z) = EX(E(Z|X = k)) = EX( ka

) = 1a

Z

V(Z) = EX(V(Z|X = k)) + VX(E(Z|X = k)) = EX( ka2

) + VX(k

a) =

=1

a2 +

1

a2 =

2

a2

X (m, 2) 1 p 2 q = 1 p

X

f(x|) = 12

e(xm)2

22 , x R

X

f(x)

f(x) = f(x| = 1)P( = 1) + f(x| = 2)P( = 2) =

= p1

21e(xm)2

221 + (1 p) 122

e(xm)2

222


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E(X) = E(E(X|)) = E(m) = mV(X) = E(V(X|)) + V(E(X|)) =

= E(2) + V(m) = 1

2p + 22(1 p) + 0 = p21 + (1 p)22

k3 k4 X(u) = lnX(u)

X X(u)

X(u)

X(u) = E(euX) =

euxf(x)dx = p

euxf(x| = 1)dx +

+ (1 p)

euxf(x| = 2)dx =

= pe21u

2

2 +mu + (1 p)e22u

2

2 +mu =

= emu[pe21u

2

2 + (1 p)e22u

2

2 ]

ex

ex =

k=0

xk

k!= 1 + x +

x2

2!+

x3

3!+ ...

X(u) = 1 + 1u + 2u2

2!+ 3

u3

3!+ ...

k X

X(u) = pe21u

2

2 +mu + (1 p)e22u

2

2 +mu =

= emu[pe21u

2

2 + (1 p)e22u

2

2 ] =

= emu[p(1 +21u

2

2+

41u4

222!+

61u6

233!+ ...) +

+ (1 p)(1 + 22u

2

2+

42u4

222!+

62u6

233!+ ...)] =

= emu

[1 +

p21 + (1

p)22

2 u2

+

p41 + (1

p)42

222! u4

+

p61 + (1

p)62

233! u6

+ ...]


52/75

X(u) = lnX(u) = mu + ln(1 +p21 + (1 p)22

2u2 +

p41 + (1 p)42222!

u4 +

+p61 + (1 p)62

233!u6 + ...) = mu + ln(1 + Y)

Y =p21 + (1 p)22

2u2 +

p41 + (1 p)42222!

u4 +p61 + (1 p)62

233!u6 + ...

ln(1 + x)

ln(1 + x) =

k=1

(1)k1xkk

= x x2

2+

x3

3 x

4

4+ ...

u3

3!u4

4!Y u3

k3 = 0

Y 12Y2u4

4!

k4 = 4!

1

222![p41 + (1 p)42]

1

2(p21 + (1 p)22

2)2

=

= 4!1

8(1 p)p(21 22)2 =

= 3p(1 p)(21 22)2

k1 = E(X) = E(E(X|)) = E(m) = mk2 = V(X) = E(V(X|)) + V(E(X|)) =

= E(2) + V(m) = 1

2p + 22(1 p) + 0 = p21 + (1 p)22k3 = 3 = E(X m)3 = 0k4 = 4 34 = E(X m)4 34 = 3p(1 p)(21 22)2


53/75

X Y a ba = b

Z = X + Y

f(z) =ab

(b a) (eaz ebz), z 0

a = 3 b = 4 Z > 8

P(Z < 10|Z > 8)

X m pY (a, r)

m pa

Xm = r p = a

a+1 X

(a, r)

X


54/75

= 0,2p = 0,01

0,005

= 5

A B 30 %A B A

A = 0,5 B B = 0,3


55/75

300 A 700 B

A BA

= 0,01 Bp = 0,001

AB

10000 A100000 B

k = 10 = 2,1

= 10

A BA B


56/75

N(m = 3, = 2) N(m = 5, = 1)

N(m = 3, 21 = 30) pN(m = 3, 22 = 100) 1 p


57/75

15 0

25 40

25

1

5(100) +

2

5(0) +

2

5(40) = 4

X

XE(X)

400 12 39212

4

XU(X) E(X)

Ux

x


58/75

U

E[U(X)] =1

5U(100) +

2

5U(0) +

2

5U(40)

E[U(Y)] =1

2U(400) +

1

2U(392)

X Y X Y E[U(X)] > E[U(Y)]X Y E[U(X)] = E[U(Y)]

X E[U(X)]

U(x) = ax + b a, b R+X E[U(X)] = aE(X) + b X

Y E[U(X)] > E[U(Y)] E(X) > E(Y)

U(x) = log(x + 400)

x > 400400 X Y

E[U(X)] =1

5log(500) +

2

5log(400) +

2

5log(360) =

=1

52,698 +

2

52,6020 +

2

52,556 = 2,6028

E[U(Y)] =1

2 log(800) +1

2 log(8) =1

2 2,9030 +1

2 0,9030 = 1,903

U(0) = log(400) = 2,6020E[U(X)] > E[U(0)] > E[U(Y)] X

Y

U(x) = x2 x 0

E[U(X)] =1

5(100)2 +

2

5(0)2 +

2

5(40)2 =

=

1

5 10000 +

2

5 0 +

2

5 1600 = 2640


59/75

E[U(Y)] =1

2(400)2 +

1

2(392)2 = 1

2160000 +

1

2153664 = 236832

Y XE[U(0)] = 0

x2 logx

r = SD(X) =

E(X E(X))2

E(X)

X

P(X = 100) =1

5, P(X = 0) =

2

5, P(X = 40) = 2

5

E(X) = 4 r1 = SD(X) = 51,22

Y

P(Y = 400) =

1

2 , P(Y = 392) =1

2


60/75

E(Y) = 4 r2 =SD(Y) = 396

= {S} X S X = U(S)P(X = U(S) a) a R

a

P(X = U(S)

a)

P(X = U(S) a) 1

a H = FX a = H(S, )

S X = U(S)

H(U(S)) = U(H(S))

a1 = H(U(S), ) X = U(S)

P(U(X) a1) ,

U

P(U(X) a1) = P(X U1

(a1)) ,


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a = U1(a1) a = H(S, )

a1 = H(U(S), ) = U(a) = U(H(S, ))

W X

P(X < W) = P(X+ W < 0)

X < W

X FX

= FX(W) = P(X < W) = P(X + W < 0)

X

FX = FX(W)

FXX N(m, )

(Wm )

FX

R(X) E(eR(X)) 1eR(W)

eR(W)


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1 E(eR(X)) =

eR(x)dFX(x) W

eR(x)dFX(x)

eR(W)W

dFX(x) = eR(W)P(X < W) = eR(W)

eR(W)

R(x)

X

X(u) = E(euX) =

euxf(x)dx

X(u) = 0

euxf(x)dx +

0euxf(x)dx

X(0) = 1 X(0) = m > 0

P(X < 0) P(X > 0) X(u) u u u

X(u) = E(X2euX) =

euxx2f(x)dx > 0, u

X(u)

u < 0 X(0) = m > 0

(0) = 1 u < 0 (u) = 1

R(x) = ux = |u|xx E(eR(x)) = E(eux

) = (u) = 1

X

eR(W) = e|u|W

X P(X < 0) P(X > 0)


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= e|u|W

u

W u

|u| = lnW

, u < 0

u

u


64/75

X(u) = lnX(u) X X(u) = eX(u)

Z(u) = e(mZ+mX)ueX(u)

W u < 0Z(u

) = 1

(mZ + mX)u + X(u) = 0

X(

|u

|)

|u| = mZ + mX

mZ =X(|u|)

|u| mX

X(u)

u

mZu mX

u

mZ =X(

|ln|W

)|ln|W

mX = I(W) mX

I(W) =X(

|ln|W

)|ln|W

W


65/75

k1 = E(X) = mX

k2 = V(X) = 2X

k3 = 3 = E(X m)3k4 = 4 34 = E(X m)4 34

X(u)u

= mX + 2X

u2!

+ 3 u2

3!+ ...

W mZ = I(W) mX

mZ = 2X

|ln|2!W

+ 3|ln|23!W2

+ ...

k Y1 + ... + YkX X(u)

Y|X = k

Y|X=k(u) = Y1+...+Yk(u) = (Yi(u))k = (Y(u))

k

k

YX

Y(u) = E(euY) = EX(E(euY|X = k)) = EX(Y|X=k(u)) == EX[Y(u)

k] = EX(eY(u)k) = X(Y(u))

Y(u) Yi

X Y(u) Y

Y

Y(u)

u=

X(Y(u))

u


66/75

a

W

X Y= Y1 + ... + YXY|X

Yi

YY(u) = X(Y(u))

X(u) = e(eu1)

X(u) = (eu 1)

a

Y(u) =a

(a u) , u < a

Y(u) = ln

a

(a u)

, u < a

Y(u) = X(Y(u)) = (eY (u) 1) = ( a

(a u) 1) = u

(a u)

Y

I(W) =

a |ln|WW > |ln|

a

= 0,05 W > 1497, 86

T tWt Xt

t Xt t


67/75

T(u)T

T(u) =T(u)

u=

Tt=1 t(u)

u

Xt

W Wtt

W

IT(W) =T(

|ln|W

)|ln|W

=T

t=1

t(|ln|W

)|ln|W

T

t=1

It(Wt)

Pt

Rt =

Pt

Pt1

Pt1


68/75

m 2

0 < < 1

P(Rt < V aR) =

P(Rt < V aR) = P( Rt m

0 k

mk

m

k = (1)k

(m + k 1)(m + k 2) (m + k k)k!


74/75

(m)! = (m)(m 1) (m (k 2))(m (k 1))(m k)!

m + k 1

k

=

(m + k 1)(m + k 2) (m + k k)k!

m + k 1

k

=

m + k 1

m 1

= 1

m + k 1 < k 0 < m < 1 k = 00! = 1


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