Stochastic process consisting both coding and proof, and some computation for constructing a transition matrix
SIMULATIONRE CALL FROM 501T -
:
Generating Random Variables from uniform 10,1 ) : -
Thin : If X is any random variable with a continuous CDF Fxln) , then
U =fn lbwif Coil ) .
(Recall : If UN Unit coil ) j Fu (u) -- ri for OEM EL) .
Pref : Since F is a CDF; F E [0,17
Cased : Assume that Fis strictly increasing between- O & I .
Fu (u ) = PLUS u }.
= Pff CX ). Eu}
= Pt Fuse Ficus } = i? { x. E FI cut }
.
E-
= F. I cus. } I U-
i . Fu (u ) = { It for
OEM EI '
U L O.
I for u 32 -
⇒ U - FCX). - llnif.com ) .
CASE : Non - doe . Fx . , Fx continuous .
Fx f. . . . . - - - -
C
I
N ,
be ,
X
Fine .
u -00,1 ) . let E > o. be such that
,
l - \
ECU & U L l - E .
÷t÷. ¥ :÷÷÷n .
This ensures that Chute] C. Cool ) Eu - E
, m) c co, I ). ]
Lice F is continuous , admin
. .
Film - O takin!IxlN=4 . - F u
, such that Fxlu , I. E ( u , Ute]
.
. I use such that Fx Cua) E Eu - E , u ) .
s I - - - - - - -
iii.i÷÷¥÷t ,a - U- E ' c (
(
o t.es Nz Ms
.
Note ; X In, ⇐ FIX) E FIND
.
i. PIX E ma} =P { FIX) E F CUD } #
For X Enz . ; then FCK) E FCUD E n. Then;
Fu cu) = PLUS on }
= Pff Cx) Eu}
3 PLFCX) E Flud } as Fleas). Rn. - IT
= PL .
X t na}
= Fx .
Cuz ) .
z n- E - ①
Similarly; X Em , # FIX). I . Flu , ! i. Pd XEN ,} = PLFCXJE Flu , ) }
.
'
. Fuku) = Ptu Eu}
= Pff CX). E u }.
I .
P { FIX) 's Flu , ) ) as netty! = Pdx tu , } = Fx In , ).
I ate - ②
Putting ① & ② together : u- E E Fu (u). E. UTE .
⇐ Ifucu). - al. E E .
teams .
i / Fufu) - ul = O.
⇐ Fula) - n . DRAB
kinky is. thai useful ? If FLXS = U N Uniflora ).
I b. Then ④= ELUL (when F- ' emits! #
i
~ Fx .
Suppose F is not continuous .
Define
[Glu ) = into { n: usfxcns. }. = min t.se : U E Fx Csc) }
.
If at Gcn) with Fxcn) tin . ⇒ f- ( Gcn ) ) = n .
A- In) = F- ' cu)
. whenever F-
' exists
.
Let UN Uniflora). Then. X - Glen). has CDF Else) .
t-xlnl.lu; I . -
g. us
.) - - - •÷Ma .T0: :- M, -- - -- - ; ' : c i 72 . VI
'
Ng i -Glad.) yea
, > Xl G- Cut.
-
-f
G-this. It ⇐ N, Glatz) - Nz
② For all n E Fez , as ). j Ma E Fxln).
i . G. (Nz? = sea .
as na.= min {MEEK,%. :
Nz Efx Cn ) } (2) For all U E (Mails ).
G- (n) : Rs .
as Nz -- min In : 'm. E FICA.}
(3) At us . ; Glu ) = Us. as us = Fxcrez ).
(4) For all n E [7,43 , '
U ,
E Fxcn).
i . G- ( Nz) -- ng
.
as Nz -- min FREER> ing] :
↳ E Fx Cns} In summary :
- In flat portions of F. i there is no increase in
prob . measure from values of a > main in the flat
segment .
•-o. Glu). = F'
' ( n't)
n 't
- If . F has jumps i
f-Ins. maps all n- ni the pimp to the
n value corresponding to Fxcn) just above
those n- ( in the jump) .
The definition of A ⇒ if u Efxcn 't)
.
for some n 't E Appoint (x) . i f- Cut ⇐six
-
i . PLXE sit }= PL flu) E at}.
Z P { U E Faint)3.
= Fx In't).
I . BERNOULLI . : - I w.pe p.
Xu Beep) ⇒ fxcn! - { o w .p. I -p . M
let U.n Uniflo, D. & . X - I ( U. ⇐ p ).
= { 1 if U Ep.
O ,
U >p.
Then Pdx- I } = PYU Ep?
=
'
fo " 1dm .
=p . .
'
. XnBu4D.
Algorithm H pseudo - code : F.
1- Generate Unthrifty).
Z - tf up E. p. i then n, := I .
else if p.cn , EI . ; then n, := O .
3- Go back to Slip I . for next sample .
Draw a graph . & - show . G- Lou) values for example U values .
I . GENERAL DISC . DISTN - ( FINITE SUPPORT) :
want to generate X , wilt mass function .
Mei } .
= pi i tei e N -
, 7¥
, pi =L .
I
Note that : Pdx- i3 = Pd . pk
.
< U .
E. II. , Pr )
.
- ti
- TT -
Xi o P
, p - -
- - -
P 1 .
Z K .
Algorithm 1- Generate lbwlhriflo, D .
2- If u Ep, , set n = I . & stop.
3 - Else . ,
set R= 2. , Sporer . Ipo & So? potp ,
@
4 - If Speer < n Eso. , set seek . & stop '
5 - Else ; set K -- Rtl
. j Speer = So I
↳ = Spurt PR-1 &
e
go back to step 4 .
III. DISC RV. ( INFINITE SUPPORT) : →
Can the loop . go on forever ? Answer : No .
Prof : let T be . the time the algorithm stops .
Then PLT > k } = PL Alg . does not stop after R
slips } = p 4. U . 7. P , t Pat - - . - t Pk
.
} = I - plate. .IE
, Pa. } - k
w = I - [ Pa. = on Z Pi
in
Them this .
MT >k3 -- thin .
( t - ?_? , pi !
"" '
R
= I - loin Z Pi k→ b- is
= I - I
= O .
Enemies simulate X. - Unit 142, . . . ,n ? Note Pf # i } - ft .
i=1 . . - in .
O ow.
Plant .
L U E In } .
= In t y 1,2. . . > n .
it In! LU E In .
then X -- y '
- -
⇒ ji c. Un . E j
⇒ X=j if j- i = Lad.
⇒ x-j if j = Lunt -11 .
⇒ X. : = LUNJ.tl. when X. w- Disc thrifty , . . . ,n?
Example : X - Geometric Cp) . Simulate from knife Coi! -
Pdx e j }. = f P C '-D ' "
j--ha . . . . . .
O o. W .
snow that x : = ItµmYI I -