Transcript
Page 1: Formula Sheet-Probability and Random Processes

Basics (Set theory, Axioms of Probability, Total Probability, Baye’s Theorem) � � �� � �� � �� � �� � �� � �� �� � �� � � � � � � � � � � �� � �� � � �� � �� � �� � �� �� � �� � 0, ������� �� � �� � ��� � ���� �� � ��

�� � … � �� � �, �������� ��|"� � #�$�%�#�%� �� � �� � ������, � & �

��� � �� � ��� � ' � �� � ��� � ��|������� � ' � ��|������� ��(|�� � ��|�(���(���|������� � ' � ��|�������

�� ) �� � ����*��� � !� � ,�!-� � � !� � ,�! ,! , � . ��& �-� ��-�0 �, �,1 �,, � . ��& �

Distribution and Density Functions (Single Random Variable) 23��� � �4 5 �� 23�∞� � 1, 23��∞� � 0, 23��� � �� �� �, � & �� ��� 8 ��� 9 4 5 �*� � 23��*� � 23���� :3��� ; &23���&�

< :3��� � 1=>= :3��� � ? (@�� � �(� 23��� � < :3��A�&�AB

>= CDEFFGDH, :3��� � 1√2KL* >�B>M�NON PQR, :3��� � S >TB, � U 0

PQR, 23��� � 1 � >TB, � U 0, V � � 1S , ��� � 1S* WDXYPGCZ, :3��� � �L* >BN*ON , 23��� � 1 � >BN*ON , � U 0, V � � L[K2 , ��� � 4 � K2 L*

EHG], :3��� � 1^ � � , � 5 � 5 ^, 23��� � � � �^ � � , �� 5 � 5 ^�, 1, �� U ^� ^GH_`GDY, :3��� � �-� �-�1��>-, V � � �, ��� � ��1 � ��

R_GFF_H, �� � ,� � :3��� � >T S-,! , V � � ��� � S CP_`PaWGb �H_ _] cPWH_EYYG aWGDYF EHaGY ]GWFa FEbbPFF�, �� � ,� � :3���� �1->�, V � � 1� , ��� � 1 � ��* ,

��|4 � �� � :B��|��:B��� ��� ��� � < ��|4 � ��:3���=>= &� 23d�4 5 ��e�4 f g�h � 2B��� � 2B�g�1 � 2B�g� , Ri] � :B�g�1 � 2B�g�

One Function of Random Variable . � 8���� � ' � 8��*� j :k�.� � :3����|8l����| � ' � :3����|8l����| 8�� :3���, 8�� 2k�.�, m � 2k>��23����

Expectations

V � � no4p � < �:3���&�=>= �� ? �4 � 4(�4(( � no4|"p � < �:3��|"�&�=

>= �� ? �4 � 4(|"�4(( �

no�4 � ^mp � �no4p � ^nomp

Linearity of Expectation

no8���p � < 8���:3���&�=>= �� ? �4 � 4(�8�4(�( �

q��o4p � L3* � no�4 � no4p�*� no4*p � �no4p�* V� � no4�p � < ��:3���&�=>=

r� � no�4 � no4p�� `DWs_t, �4 U g� 5 no4pg bZPuXbZPt, �|4 � no4p| U v� 5 L*v* w3�x� � no yz3p � < yzB:3���&�=>=

��� � no |3p � < |B:3���&�=>=

Φ}~� ������~����0� � no4�p � V� :3��� � 12K < >yzBw3�x�&�=>=

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 � < :3kd4��, x�, m��, x�h=

>= &x=>= ��� 9 4 5 �*, .� 9 m 5 .*� � 2��*, .*� � 2���, .*� � 2��*, .�� � 2���, .�� :��, .�� �*2��, .����.

2��, .�� < < :��, .�&�&.�>=

B>= �HiPR: :��, .� � :3���:k�.�, 2��, .� � 23���2k�.� �4 f �, m f .� � 1 � 23��� � 2k��� � 2��, .�

� � C��, �� j 2���� � < < :��, .�&�&.B��,��>B��,����>=

������=�0>������> =

no�4 � ^mp � �no4p � ^nomp no4mp � no4pnomp �� �� � ����

� & � j � ����, � ����� � & �

� � C��, ��, � � Z��, �� j 2��, x� � � :3k��, .�� &�&. j :��, x� � :3kd4��, x�, m��, x�h �����, x��� ����, x��x�.��, x��� �.��, x��x � � :3k�4��, x�, m��, x���8��, .��� �8��, .��.����, .��� ����, .��. �

�PGuHGa�: 2���� � � :��, ��&����� ��� j :� ��� � ¡ ¢£ ���¡� � ¡����¡� :�^���, �� � ¡ ���¡� :�����, �� � � ¤¥�B,��¤� &.���� ��� w�x�, x*�� w3�x��wk�x*�, � & � �3k � n¦�4 � §B�dm � §�h¨� no4mp � no4pnomp ©B� � �B�LBL� w�x�, x*� � < < :��, .� y�zªB«zN��&�&.=

>==

>= < < :��, .�&�&. � 1=>=

=>=

:��|.� � :�.|��:���:�.� � :�.|��:���� :�.|��:���&�=>= w3k�x�, x*� � no yzª3 yzNkp� � yzªB«yzN�:3k��, .�&�&.�

:3k��, .� � 14K* � �>yzªB«yzN��w3k�x� , x*�&�&��

Page 2: Formula Sheet-Probability and Random Processes

Φ���, �*� � no |ª3«|Nkp � � |ªB«|N�:3k��, .�&�� :k�.|4 � �� � :3k��, .�:3��� n�8�4�|"� � < 8���:��|"�&�=>=

nomp � n3onkom|4pp

n¬no8�4, m�|4p­ � no8�4, m�p no8��4�8*�m�|�p � no8����8*�m�|�p no®���4 � no4p� � �m � nomp�¯*p U 0 :��, .|�� � :3k���, ., ��:����

Sequences of Random Variables °± ²Fa. uX b_HFaDHa, � no�m � ��*p, �� min x� & &� � 0j � � nomp

�GHPDW °± ²Fa, � no®�m � ��4 � ^�¯*p, �� min x� � �� � 0 � & � �^ � 0 j � � �3kL3* , ^� rk � �3kL3* r3 4 � 4� � ' � 4�j 23ª,…,3·���, … , ���

2���, �¸� � 2���, ∞, �¸, ∞� :���, �¸� � < < :���, �*, �¸, �¹�&�*&�¹=>=

=>=

º���^�� " ���& � & � & � .��»� � ����»� 5 ' .��»� � ����»� j ,¼½��& � ��������� j :-�.� � !�, � 1�! � � ,�! 2B->��.�o1 � 2B�.�p�>-:B�.� 4, m � & � � & ¾ �4 � m j L�*=L3* � Lk*

Stochastic Convergence 4� PtPWX¿ZPWPÀÁÁÁÁÁÁÁ 4, �� à ∞ j |4� � 4| 9 v, :�� ��� f Ä 4� DY`_Fa PtPWX¿ZPWP �_W ¿GaZ Å0Æ�ÀÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁ 4, �� à ∞ j ®|4� � 4| 9 v¯ � 1 4� °± ±PHFPÀÁÁÁÁÁ 4, �� à ∞ j no|4� � 4|*p à 0, �� à ∞ 4� ÅW_uDuGYGaXÀÁÁÁÁÁÁÁ 4, �� à ∞ j ®|4� � 4| f v¯ à 0, �� à ∞ 4� ÇGFaWGuEaG_HÀÁÁÁÁÁÁÁÁ 4, �� à ∞ j 2���� à 23���, �� à ∞ 4� ÇPHFGaXÀÁÁÁÁ 4, �� à ∞ j :���� à :3���, �� à ∞ "� j ��^

4�ÈÈÈÈ � 1 �4� � ' � 4��, x� � 4(�� ��& x��� V � r, ¿PDs YD¿ j4È� ÅW_uDuGYGaXÀÁÁÁÁÁÁÁ r, FaW_HC YD¿, j4È� DPà r, �� à ∞

bPHaWDY YG`Ga, ¾� � 1 �4� � ' � 4��, x��� V � � r, ��� � L* 9 ∞ j ¾� à ¾~Ê�r, L*�

Stochastic Processes HaZ _WiPW , 2��aÆ�…��aH���Æ, … , ��; �Æ, … , ��� � ®4���� 5 ��, … , 4���� 5 ���, :��aÆ�…��aH���Æ, … , ��; �Æ, … , ��� � ¤·¢�daÆh…��aH��BÆ,…,B·;¼Æ,…,¼·�¤Bª…¤B· 2���; ��� � 2���, ∞; ��, �*� :���; ��� � < :���, �*; ��, �*�=>= &�* r3��� � n�4���p � < :��; ��&�=

>= no4*���p � ��, ��

DEa_b_WW, ���, �*� � no4����4��*�p � < < ���*:���, �*; ��, �*�&���*=>=

=>=

DEa_b_tDW, ����, �*� � ���, �*� � r3����r3��*�

b_WW b_P]], ����, �*� � ����, �*�Í����,������*,�*� bW_FF b_WW, Ì3k���, �*� � no4����m��*�p bW_FF b_tDW, ����, �*�� Ì3k���, �*�� r3����rk��*� ¿ZGaP H_GFP, ������, �*� � 0, �� � �*, r � 0, ��Î� � 1@�Î� ±±±, :��aÆ�…��aH���Æ, … , ��; �Æ, … , ��� � :��aÆ�…��aH���Æ, … , ��; �Æ � b, … , �� � �� �±±, V � � �� �, no4����4��*�p � Ì��� � �*� � Ì�Î�, no4*���p � Ì�0�, ��Î� � Ì�Î� � |r|*, ��Î� � ��Î���0� , Ì�Î� � Ì��Î�, Ì�0� U Ì�Î� Ï_GHaYX �±±, no4����4��*�p � Ì3k��� � �*� � Ì3k�Î�, �3k�Î� � Ì3k�Î� � r3rk

Systems with Stochastic Inputs m��� � ������, 4��� � � ��� `P`_WXYPFF, m����& � &� � 4���) � ��� �� V V��.� �� �� ��� j ������ ���� ��� YGHPDW, m��� � Ðo4���p � 4��� Ñ ���� � < 4�� � g���g�&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���, �* � g���g�&g=>= , Ìkk �? R_¿PW FRPb, ��Õ� � Ö�Ì�Î�� < Ì�Î� >y×Ø&Î=

>= j Ì�Î� � 12K < ��Õ� y×¼&Õ=>= Ù�Ú� Û ±�Ü�, @�Î� Û 1,1 Û 2K@�Õ�, yÝØ Û 2K@�Õ � Þ�

Integration (all integrals are w.r.t x) < ß B � 1ß�  B < 1� � � |�| < ß �B«�� � 1ß� >�B«�� < �� � ��«� � 1 < sin��� � �cos ��� < cos��� � �� ���

< tan��� � �ln |������| < ß� B«�� � ß 1ß� �ß B«�� < 1�  � 1�� � 1 �> «� < ��l � �� � < �A�

Differentiation &&� ���� � ��l � ��A &&� æ��ç � ��l � ��A�* &.&� � &.&� &�&�

&&� ���� � ��>� &&� � ߠB� � � ߠB

&&� æ  Bç � � ��*  B &&� �sin ���� � cos ��� &&� �cos ���� � ��� ���

Matrix Operations è� ^� &è � �& � ^� �>� � 1|�| é��� �*���* �**ê , �(y�� � � �: � � ë

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)


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