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 -