Download Continuous Random Variables: A Basic Introduction and more Lecture notes Probability and Stochastic Processes in PDF only on Docsity!
4. Continuous Random Variables
- (^) I. Continuous (^) Sample Space
PlX=x] -^ -^ o^ TaeSx
4.2 . CDF , Fxlxl = PIX Ex^ )
(a) Fx^ I^ -001=0^ (b)^ Fxloo)^ -^ I^ (c)^ PluckExz^ ]^ =^ Fxfkz)^
4. 3.^ PDF^ (Probability Density function^ )
fxlx )^ =^ dT doc (a) (^) fun Zo^ (b) (^) Fxtktfjofxluldu cc, (^) f%fh4dx= I →
place xD^
= find >c^ '^ I' I
- (^) Expected Values ( ¥¥¥¥ )^ Jae
" du K - - TN
④ ④^ =^ xe
" (^) -
e "^ t C →^ pklkt-pxlk.tl KEK^ )
EH (^) ] (^) =/ !! xffxldx aye " = (^) Fxlk) - Fxlkt ) I e " ELGIN]^ =L! glxsfh da OT e
" =^ - It e
" I - ( I^ -^ e-^ ''^ " '^ "
x (^) = e- dlkl^ )^ I I^ - e
4.5 . Families of continuous random variables
( a. b) bta Uniform RV^. ftp..gl/lb-a1,aExsb
EH ]^ -
so , elw^
Vfx ) = ( b^ -^ a)^412
* Exponential RV^.
" "
fun ,^ =
g doe " " ,
KZO Efx ) = Ya
ceiling geometric ex ) Ekta 'zH2t^ , elw^ VIX^ )^ = IN Ex 4.11 RV^.
* Erlang RV .
f (^) ,
ftp.azo
Efx ,^ =^ %
with A te
" ex (^) ) meet (^) Ekta '3HH^ Vfx) - Man A I^ →^ exponential Rv^. ° (^) ,^ elw
Pf )^ exponential
RV .
ECM = I? x^ de " " die = fixed "
= ( o - o ) - ( o^ -
⇒
I VIA (^) =/! Tae ' "dx=fYet¥^ " (^) - Ela et " (^) - Eet " ) ! = to
the
CDF = flood e-^ tu du -^ - f-e- tu ) ! = I - e^ "^ " , azo { (^) o , elw
- (^6).^ Gaussian^
Random Variables
h¥
e-
( " MMM
→ X ~^ N^ I^ μ , r ) :^ fxlx, =^ -
- Xr^ Nfu , o ) →^ Y^ =^ a^ Xtb^ -^ N^ (^ qutb , la^ It^ )
* standard normal^ RV^.
CDF :^ EH . e
. Eda "
Finer (^) EHF) place E^ b)^ = Platts II^ Itbye) = EPI ) - E (^) ( ⇒
standard Normal^ Complementary CDF
QA ) = PLZ >^ z^ ) =^ I^ -^ Eez)
① Jo
" ( aiftbx^ ) da^
② f Lot = o Zo
③ fu ) =^ atb^ I o
in :^
② aco
0¥
' ⇐^ * :^ b^ z^ -^ a
i. ' gatlzb 't ,
ELY ] =^ Po
MY >^ Po]^
= Jp? ¥^
e- Npo^
dy = (^) I - Fy (^) ( Po ) = I - { I^
- e
- Ei Poy = e - I fycy)^ = f I e
- & , y^
Zo
O , elw II
Erez
's e-^ Eg
Qcrzy fylyl
