1- Parameter Families of Random Variables

we will often have occasion to considert - parameter families of random variables

{ Xp : te la,b) ) - es sa - b Sos

we will study many stochastic processesof this form ,

but they also come up more

innocuously .

Eg .

Let X be a random variable in Crisp ) .

Then etx 30 for all te IR,so

14×111 #letx ) = fetxdp exists in ↳ a ]

Note : if Hk e then txsltxlselxl

If Fe > o sk ed" e Lt i. fetxdlpsfe""DIP -

- Ile" "Yes .

i. CHEL' ft E. C.-GE) .

Prop .16.30 Let crit

,g) be a measure space ,and f : Ca

,b) xr → IR St

.

t.

w↳ fit,

w) is measurable for each tha,b )

2. fl to

,

. ) E Lt Crisp) for some to e- Ca,

b )

3.

2 flat Usw ) exists for m - a.e.

w and for every becqb)4. There is gehtCrim) sit . 1¥, lt.ws/sguo)for µ - a

-e - w and for every te Ca

,b) -

Then fit; ) c- L' for all to Ca

,b)

,

ti-ffct.co)Mdw) is differentiable en la,b),

and dq.ffct.wsMdw) = fHgtgIµ Idw) -

Pf . tgfqllsw) = LIE n - ( fifth , w) - fugw)) between thet

Mean Value Thm : lflt.wf.f.fct.in/=lffqCt9wJ/sgcw)i. Hans it!.ws .ci#lgupj.iflts:Y ,

"

" ".

Define GH) - fflt.wsMdw) .

Then

GHI - GH = f fairy-fit;D

t - to t.fm/dw)

t.m.am#I=fisi.f ""

IH ftp.n.ll-pvdw)by DCT . H

Eg .

Let fit,w ) = etxlw)

.

measurable ft V

If RM t Lt,then so is ft; ) for Gay ) te -- f

¥.

. dateTX = xetx exists ft t

Let attest ) Then 1×1 sea""

i. IX et Xt s PEK ! e

" - 04×1=61×1

-:# Is e""

ft msn.ae

"

ft!!" -ne

tht

i. at , Itt se.

-

'

- by deal day My = dat # let'T = # Hetty = #Hetty

Repeat : ¥ynµ×µ ) = #(xn et 'T fo ltke .

i. Mx'"

co) = LENI .⇐ moments of X .

Actually,we can do this all at once . Note : etx.no?tnqXh .

Prep : If e""t Lt then Mx is analytic on te, e ) and

Mxlt) - Ia tf.

TECH la se.

Pf. Let fnlt.ws .- Eat

.

Hugh

-

'

- I falls as I s Intl Nwsl"f Einen

.

I NP = e'" 'as et

.

fo Lt.

. ) → etx one

i. by DCT ⑤ Half .) f → El et'll aMx H)

"

El! in = E. In.

#oxy .

i . as > Ef et'T = ftp./" " ) - E. tf #KY .

CautionIt's really important that the conditions of the

differentiate- under - an - integral propositionheld as

. independently of t .

Eg . Lebesgue measure ( cell,Basil? X)

fltw) = Dust = Degli Iwl .

1-.

For fixed t, Gee

,× ,V measurable .

2. If It,

-H Sl and Hegel) ekes so fllg.de L'

3- Film '- Itf 's . 138 }

-

- { §' fgt!:

.

.

- o as .

4. gzo C- Lt . V

daftest , di = ddtffctwjlldw) = 1¥, DX = fodteo .

= days ↳ H -

- deft -1.

*.