Formula Sheet-Probability and Random Processes

Basics (Set theory, Axioms of Probability, Total Probability, Baye’s Theorem)
     
   
  
  
 
  
 
 


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  ! !  ,  . &  -    ,!   ,! ,! Distribution and Density Functions (Single Random Variable) 23   4 5  23 ∞  1, 23 ∞  0, 23     ,  &   8


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 *  =

< :3   1

 -



:3   ? ( @  ( 

B

23   < :3 A&A

-0 ,, 1

,

,  . & 

&23  & PQR, :3   S >TB ,  U 0

 9 4 5 *   23 *   23  

CDEFFGDH, :3  

1

B>MN > ON

:3  ;

>= √2KL * N >B >B N 1 1  K 4K * PQR, 23   1  >TB ,  U 0, V   ,   * WDXYPGCZ, :3   * *ON , 23   1  *ON ,  U 0, V   L[ ,   L S S L 2 2 1  ^GH_`GDY, :3   - - 1>- , V   ,   1   EHG], :3   ,  5  5 ^, 23   ,  5  5 ^, 1,  U ^ ^ ^ CP_`PaWGb H_ _] cPWH_EYYG aWGDYF EHaGY ]GWFa FEbbPFF,   ,  :3  S R_GFF_H,   ,  :3   >T , V     S 1 1 ,!  1 -> , V   ,   ,  * = 2B   2B g :B g :B |
 23 d4 5 e4 f gh  , Ri] 


|4   


 
  < 
|4  :3  & 1  2 g 1  2B g :B  B >= One Function of Random Variable :3   :3   8 :3 , 8 2k ., m  2k> 23  .  8   '  8*  j :k .  l  ' l |8  | |8  | Expectations = = no4  ^mp  no4p  ^nomp Linearity of Expectation V   no4p  < :3 &  ? 4  4( 4(  no4|"p  < :3 |"&  ? 4  4( |"4(  >=

=

(

>=

no8p  < 8:3 &  ? 4  4( 84(  (

>=

no4p `DWs_t, 4 U g 5 g

r  no4  no4p

=

Φ  no |3 p  < |B :3 & >=

(

>=

qo4p  L3*  no4  no4p*  no4 * p  no4p*

L* bZPuXbZPt, |4  no4p| U v 5 * v

Φ}~ €‚ƒ„…~ƒ„ 0  no4  p  V

=

V  no4

p

=

 <   :3 & =

>=

w3 x  no yz3 p  < yzB :3 & >=

1 :3   < >yzB w3 x& 2K >=

Joint Distribution, Density and other relations(Functions of two random variables, Two functions of two random variables) = = = 23k ∞, .  23k , ∞=0 23   23k , ∞  :3   < :, .&. :† ‡  < :†ˆ ‡, x&x  < :3k d4‡, x, m‡, xh &x >=

 9 4 5 * , . 9 m 5 .*   2* , .*   2 , .*   2* , .   2 , .  ‹HiPR: :, .  :3 :k ., 2, .  23 2k .   CŽ,  j 2 ‡  <

Š‘’=

<

BŠ,

Š0>Š‘’> = >BŠ,‘’>=

:, .&&.

>=

>=

:, . 2, . B Š ‰ * 2, .  < < :, .&&.  ‰‰. >= >=

4 f , m f .  1  23   2kŠ  2, . no4  ^mp  no4p  ^nomp ” &  j • , •  no4mp “ no4pnomp     – ” & 

‰‡, x   CŽ, , —  ZŽ,  j 2‡, x  ˜ :3k , . &&. j :‡, x  :3k d4‡, x, m‡, xh š ‰‡ ‰.‡, x ™ ‰‡

œPGuHGa: 2 ‡  ž  :, ‡& j : ‡  Ÿ

¡ ¢£  ¡



¡Ÿ ¡

‰‡, x ‰x š  :3k 4‡, x, m‡, x ‰.‡, x ‰8, . ‰8, . ‰ ‰. ‰x š š ‰›, . ‰›, . ‰ ‰. Ÿ ¤¥B,Š ¡  wx , x*  :^‡, ‡  :‡, ‡  ž  &. ¡ ¤  w3 x wk x* ,  & 

= = = = BŠ 3k  n¦4  §B dm  §Š h¨ ©BŠ  wx , x*   < < :, . yzªB«zNŠ &&. < < :, .&&.  1 LB LŠ  no4mp  no4pnomp >= >= >= >= :.|: :.|: w3k x , x*   no yzª 3 yzN k p 1 :|.   = :3k , .  * ˜ >yzª B«yzNŠ w3k x , x* && :. 4K ž>= :.|:& yzª B«yzN Š ™ , ˜ :3k .&&. ™

= :3k , . n84|"  < 8:|"& :3  >= no8 48* m|p  no8 8* m|p no®4  no4p  m  nomp¯* p U 0

:k .|4   

Φ , *   no |ª3«|Nk p  ˜ |ªB«|NŠ :3k , .& ™

n¬no84, m|4p­  no84, mp

Sequences of Random Variables °± ²Fa. uX b_HFaDHa, &  nom  * p,  min x› 0 & j   nomp

œGHPDW °± ²Fa,  no®m  4  ^¯* p,  min x›  rk 

2 , ¸   2 , ∞, ¸ , ∞

4  4  '  4 j 23ª,…,3·  , … ,  

3k r L3* 3

=

=

nomp  n3 onk om|4pp

:, .|‡ 

‰ ‰ 3k  0 &  0 j   * ,^ ‰ ‰^ L3

º^ " ›& ” &  &  4, m  &   & ¾  4m j L†* =L3*  Lk*

: , ¸   < < : , * , ¸ , ¹ &* &¹ >= >=

! . »  ’ » 5 ' . »  ’ » j , &   j :- .  2 -> .o1  2B .p>- :B . ,  1!   ,! B PtPWX¿ZPWP

¼½

à ∞ j |4  4| 9 v, : 

Stochastic Convergence 4 ÀÁÁÁÁÁÁÁÂ 4,  °± ±PHFP

4 ÀÁÁÁÁÁÂ 4,  ÇGFaWGuEaG_H

à ∞ j no|4  4| p à 0,  *

f

̰

Ä

DY`_Fa PtPWX¿ZPWP _W ¿GaZ Å0Æ

4 ÀÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÂ 4, 

ÅW_uDuGYGaX

4 ÀÁÁÁÁÁÁÁÂ 4,  ÇPHFGaX

à ∞ j ®|4  4| 9 v¯  1

à ∞ j ®|4  4| f v¯ à 0, 

̰

" j ^

4 ÀÁÁÁÁ 4,  à ∞ j :  à :3 ,  à ∞ 4 ÀÁÁÁÁÁÁÁÁ 4,  à ∞ j 2  à 23 ,  à ∞ ÅW_uDuGYGaX DP 1 È4ÈÈÈ  4  '  4 , x›  4(  & x› V  r, ¿PDs YD¿ j4È ÀÁÁÁÁÁÁÁ r, FaW_HC YD¿, j4È à r,  à ∞ bPHaWDY YG`Ga, ¾  Stochastic Processes

1

4  '  4 , x› V   r,   L * 9 ∞ j ¾ à ¾~Êr, L * 

HaZ _WiPW , 2ŽaÆ …ŽaH  Æ , … ,  ; Æ , … ,    ®4  5  , … , 4  5  , :ŽaÆ …ŽaH  Æ , … ,  ; Æ , … ,    2 ;    2 , ∞;  , * 

=

: ;    < : , * ;  , *  &* >= =

=

 , * 

>= >=

=

r3   n4p  < :; &

DEa_b_WW, Ì , *   no4 4* p  < <  * : , * ;  , * & *

:3k† , ., ‡ :† ‡

>=

¤· ¢Žda h…Ža  BÆ ,…,B· ;¼Æ ,…,¼· H Æ ¤Bª …¤B·

no4 * p  Ì, 

DEa_b_tDW,  , *   Ì , *   r3  r3 * 

bW_FF b_tDW,  , *   Ì3k  , *  Í,  *, *   r3  rk *  ¿ZGaP H_GFP, ˆˆ  , *   0,  “ * , r  0, Î  1@Î ±±±, :ŽaÆ …ŽaH  Æ , … ,  ; Æ , … ,    :ŽaÆ …ŽaH  Æ , … ,  ; Æ  b, … ,    Î —±±, V    , no4 4* p  Ì  *   ÌÎ, no4 * p  Ì0, Î  ÌÎ  |r|* , Î  , ÌÎ  ÌÎ, Ì0 U ÌÎ 0 Ï_GHaYX —±±, no4 4* p  Ì3k   *   Ì3k Î, 3k Î  Ì3k Î  r3 rk Systems with Stochastic Inputs m  , 4    `P`_WXYPFF, m &  &  4 )    V V.    j    b_WW b_P]],  , *  

bW_FF b_WW, Ì3k  , *   no4 m* p

=

YGHPDW, m  Ðo4p  4 Ñ ›  < 4  g›g&g, x›  ›  Ðo@p, n¬Ðo4p­  Ьno4p­, 4  m   >=

=

YGHPDW, 4Ò › mÒ, œÓ‹, 48 m8 , YGHPDW, nomp  no4p Ñ ›  r3 < ›g&g =

YGHPDW, Ì3k  , *   < Ì33  , *  g›g&g , Ìkk ? >=

=

R_¿PW FRPb, Õ  ÖÌÎ < ÌÎ >=

>y×Ø

>=

&Î j ÌÎ 

1 = < Õ y×¼ &Õ 2K >=

ÙÚ Û ±Ü, @Î Û 1,1 Û 2K@Õ, yÝØ Û 2K@Õ  Þ Integration (all integrals are w.r.t x) 1  B 1 1 >B«Š  « < sin  cos  < cos    < ß B 

<   || < ß B«Š 

<   ß  ß 1 1 ß B«Ÿ 1 1 < tan  ln || < ß B«Ÿ  ß <  l    < A

<    > « ß    1 Differentiation   & &. &. & &  & ß B &    &  l  A > ß B Bç      l  A         ß æ

B æ ç & & & & & & & * &  * & & sin   cos  cos     & & Matrix Operations 1  *  ^ > è è  &  ^
 é ê ,     :   ë  & |
| * ** (y Prepared by: Hammad Munawar (Institute of Avionics and Aeronautics, Islamabad, Pakistan) [email protected]

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