formula sheet-probability and random processes

2
Basics (Set theory, Axioms of Probability, Total Probability, Baye’s Theorem) 0, , | , | | | | | | ! ! ! ! ! , , Distribution and Density Functions (Single Random Variable) 1, 0, , 1 , 1 2 , , 0 , 1 , 0, 1 , 1 , , 1 , 0, 2 , 4 2 , 1 , , , , 1, , , , 1 , ! , , , 1 , 1 , | | | 1 , 1 One Function of Random Variable | | | | , , Expectations | | | Linearity of Expectation , , | | Φ Φ 0 1 2 Joint Distribution, Density and other relations(Functions of two random variables, Two functions of two random variables) , , =0 , , , , , , , , , , , , , , , : , , , , 1 , , , , , , , , , , , , , , , , , , , , , , , , , , : , , , , , , , , , 1 | | | | , , , 1 4 ,

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A simple formula sheet that lists basic and essential formulas for a first course in Probability and Random processes. It can also be used in examinations that allow students to bring only one A4 sized formula sheet.Contains basics of set theory, axioms of probability, probability distribution functions, stochastic convergence and a lot more.Can be edited or distributed freely, as long as the name of the original author is included in the document.If you like it, or dont like it, please drop in a comment.

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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)