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
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&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 = = no 4 ^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 & >=
(
>=
q o4p L3* no4 no4p* no4 * p no4p*
L* bZPuXbZPt, |4 no4p| U v 5 * v
Φ}~
~ 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 < :, .&. : < : , x&x < :3k d4, x, m, xh &x >=
9 4 5 * , . 9 m 5 .* 2* , .* 2 , .* 2* , . 2 , . HiPR: :, . :3 :k ., 2, . 23 2k . C, j 2 <
=
<
B,
0>> = >B,>=
:, .&&.
>=
>=
:, . 2, . B * 2, . < < :, .&&. . >= >=
4 f , m f . 1 23 2k 2, . no 4 ^mp no4p ^nomp & j , no4mp no4pnomp &
, x C, , Z, j 2, x :3k , . &&. j :, x :3k d4, x, m, xh ., x
PGuHGa: 2 :, & j :
¡ ¢£ ¡
¡ ¡
, x x :3k 4, x, m, x ., x 8, . 8, . . x , . , . . ¤¥B, ¡ wx , x* :^, : , &. ¡ ¤ w3 x wk x* , &
= = = = B 3k n¦4 §B dm § h¨ ©B wx , x* < < :, . yzªB«zN &&. < < :, .&&. 1 LB L no4mp no4pnomp >= >= >= >= :.|: :.|: w3k x , x* no yzª 3 yzN k p 1 :|. = :3k , . * >yzª B«yzN w3k x , x* && :. 4K >= :.|:& yzª B«yzN , :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«|N :3k , .&
n¬no84, m|4p no84, mp
Sequences of Random Variables °± ²Fa. uX b_HFaDHa, & nom * p, min x 0 & j nomp
GHPDW °± 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 ÀÁÁÁÁÂ 4, Ã ∞ j : Ã :3 , Ã ∞ 4 ÀÁÁÁÁÁÁÁÁÂ 4, Ã ∞ j 2 Ã 23 , Ã ∞ ÅW_uDuGYGaX DP 1 È4ÈÈÈ 4 ' 4 , x 4( & x V r, ¿PDs YD¿ j4È ÀÁÁÁÁÁÁÁÂ r, FaW_HC YD¿, j4È Ã r, Ã ∞ bPHaWDY YG`Ga, ¾ Stochastic Processes
1
4 ' 4 , x V r, L * 9 ∞ j ¾ Ã ¾~Êr, L *
HaZ _WiPW , 2aÆ …aH Æ , … , ; Æ , … , ®4 5 , … , 4 5 , :aÆ …aH Æ , … , ; Æ , … , 2 ; 2 , ∞; , *
=
: ; < : , * ; , * &* >= =
=
, *
>= >=
=
r3 n4 p < :; &
DEa_b_WW, Ì , * no4 4 * p < < * : , * ; , * & *
:3k , ., :
>=
¤· ¢da h…a 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, , * 0, * , r 0, Î 1@Î ±±±, :aÆ …aH Æ , … , ; Æ , … , :aÆ …aH 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 Ðo4 p 4 Ñ < 4 gg&g, x Ðo@ p, n¬Ðo4 p Ьno4 p, 4 m >=
=
YGHPDW, 4 Ò m Ò, Ó, 4 8 m 8 , YGHPDW, nom p no4 p Ñ r3 < g&g =
YGHPDW, Ì3k , * < Ì33 , * gg&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« « < sin cos < cos < ß B
< || < ß B«
< ß ß 1 1 ß B« 1 1 < tan ln | | < ß B« 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]
(COPYRIGHT STATEMENT: May be used / distributed / edited freely, as long as the name of the original author is included)
