Samenvatting Stochastics and Simulation for Finance (2DF20)

, Chapter s




Dixrete random variables

The Random variable X 83
range ag 90




:
s1....
with IP
Pr
=


=(X=K)
IELXJ -9 mean
E:okPK
lECX : KPr




.-Ekpj
var CX
= variance
3
3-IECX3
Ęo
Binonial
distribution distribation I independent Bernouilli Erials with
og number of successes in
: n successes p




.
Benoted bag n Bin
(rips
IPIXK PY CI kEEO 1 ... n PELO
-PjMY
,
,




3
,13
)=(Î)
lECX =
mp
3
VarCxJ = APC
1-P)
Poisson aistribation denoted by Pois (
x)
:
IPLX 03, X
30
1
s....
=K)=E*YkEEO,
IECX = and War LKJ =
x
I
3
o


Geometric distribution number of success before first foil
':
denoted by Xn beolp
)
IPIX = ( hhoss -..03, PECos
=K)
"
F
FlpI
r-plpR,
IECXJ ard Var CX crupp
:
3=
- Memergless propertg PPLXS KIMIXIN
=IPLXEKS
does depent
s
not on n
:
Binomial theoren Ywo ()
Y CatbJr
a"gN-ke
:
Taglor serves : et RoX
2?
:
If X is value then IECX o IPCXsK
3:8
a non
-
)
regative
probability generating functions
The E )
probability sencrating function og
a random variaable X with
range .
...1 8 and
probability distribution
0j1
Pr for K 1 . .
189 is
defined by
I.
=IPLX=K)
=O1
s
gI = ĘOPaZY IELZTJ
(2):
PGF og distribution 9 (2) "
x
geometric
Fpz
a =
:
à (2- 1)
PGF og : distribution 9 12) a
x
poisso
an
: