Transcript
Page 1: Variables aleatorias reales

&RQ�IUHFXHQFLD �KD\�LQWHUpV�SRU� ORV�YDORUHV�QXPpULFRV�TXH�VH�SXHGDQ�GHGXFLU�GH� XQ� H[SHULPHQWR� DOHDWRULR�� JHQHUDOPHQWH � GLFKRV� YDORUHV� FRUUHVSRQGHQ�D� ORV�GH�XQD�YDULDEOH�DOHDWRULD��(O�HVStULWX�GH� ODV�YDULDEOHV�DOHDWRULDV�HV�HO�GH�WUDQVIRUPDU�ORV�HOHPHQWRV�GH�XQ�HVSDFLR�GH�SUREDELOLGDG�HQ�Q~PHURV�UHDOHV��(Q�HVWH�FDStWXOR�VH�SUHVHQWDQ�ORV�FRQFHSWRV�GH�YDULDEOH�DOHDWRULD�UHDO��IXQFLyQ�GH�SUREDELOLGDG��GLVWULEXFLyQ�GH�SUREDELOLGDG��YDORU�HVSHUDGR��YDULDQ]D��GHVYLDFLyQ�HVWiQGDU�\�PRPHQWRV�SDUD�YDULDEOHV�DOHDWRULDV�WDQWR�GLVFUHWDV�FRPR�FRQWLQXDV�

�����&RQFHSWR�GH�YDULDEOH�DOHDWRULD�

(Q�HVWH�DSDUWDGR�VH�GHILQH�ODV�YDULDEOHV�DOHDWRULDV�UHDOHV�\�VH�SUHVHQWDQ�DOJXQRV�HMHPSORV�� 6HD� � XQ� HVSDFLR� GH� SUREDELOLGDG�� GRQGH� � HV� HO� HVSDFLR�PXHVWUDO�R�FRQMXQWR�GH�SRVLEOHV�UHVXOWDGRV�GH�XQ�H[SHULPHQWR�DOHDWRULR�� �HV�XQ� iOJHEUD�R�IDPLOLD�GH�VXEFRQMXQWRV�GHO�HVSDFLR�PXHVWUDO�\�3 �XQD�PHGLGD�GH�SUREDELOLGDG�GHILQLGD�VREUH� �

6HJ~Q�FRQ�6KDR���������XQD�YDULDEOH�DOHDWRULD�UHDO�HV�XQD�DSOLFDFLyQ�

�; 5

7DO�TXH�SDUD�WRGR�HYHQWR�( �HQ�HO� iOJHEUD�GH�%RUHO�VH�WLHQH�TXH

�� �; (

(Q�HVWH�FRQWH[WR��OD�YDULDEOH�DOHDWRULD� ; �HV�XQD�IXQFLyQ�TXH�WLHQH�OD�SURSLHGDG�GH�VHU�XQD�IXQFLyQ�PHGLEOH��(Q�ORV�HMHPSORV�TXH�VH�SURSRUFLRQDUiQ�D�OR�ODUJR�GHO�FDStWXOR�VH�FRQVLGHUDUi�HO�VLJPD�iOJHEUD�FRQIRUPDGD�SRU�HO�FRQMXQWR�GH�SDUWHV�GHO�HVSDFLR�PXHVWUDO�

Page 2: Variables aleatorias reales

(MHPSOR������&RQVLGHUDU�ORV�UHVXOWDGRV�SRVLEOHV�DO�LQWHQWDU�HVWDEOHFHU�VL�HO�MXJDGRU�5�FRQYHUWLUi�R�QR�HO�ODQ]DPLHQWR�TXH�KDUi�GHVGH�HO�SXQWR�GH�SHQDOL]DFLyQ�HQ�XQ�SDUWLGR� GH� I~WERO�� (O� HVSDFLR�PXHVWUDO� HVWDUi� FRQIRUPDGR� SRU� GRV� UHVXOWDGRV�SRVLEOHV�� F FRQYLHUWH��Q QR�FRQYLHUWH��HV�GHFLU�

6H�GHILQH�OD�VLJXLHQWH�YDULDEOH�DOHDWRULD�

�; 5

���

F

Q

(Q�HVWH�FDVR�VH�WLHQH�TXH� � � �; Q �\�� � � �; F �� ; �HV�XQD�YDULDEOH�DOHDWRULD�FX\R� UDQJR�HV�XQ�FRQMXQWR� ILQLWR�GDGR�SRU� ��(Q�HVWH� FRQWH[WR� ;5GHQRWD�HO�UDQJR�GH�OD�YDULDEOH�DOHDWRULD�PHQFLRQDGD�

(MHPSOR����� 3DUD�HO�HVSDFLR�PXHVWUDO�FRQVWLWXLGR�SRU�ORV�VLJXLHQWHV�UHVXOWDGRV�

'RQGH� G ��DUWtFXOR�GHIHFWXRVR�� Q ��DUWtFXOR�QR�GHIHFWXRVR��VH�GHILQH�OD�VLJXLHQWH�YDULDEOH�DOHDWRULD�

�; 5 �

; ��Q~PHUR�GH�DUWtFXORV�GHIHFWXRVRV�HQ�FDGD�UHVXOWDGR�GH� �

� � �; GG �

� � � � �; GQ ; QG �

� � �; QQ �

/D�YDULDEOH� ; �WRPD�ORV�YDORUHV�HQWHURV����������(V�GHFLU�� ��

�����9DULDEOH�DOHDWRULD�GLVFUHWD�

6HD� ; �XQD�YDULDEOH�DOHDWRULD�GHILQLGD�VREUH�XQ�HVSDFLR�PXHVWUDO�� �\� ;5 �VX�UDQJR�HQ�HO�FRQMXQWR�GH�ORV�Q~PHURV�UHDOHV� 5 ��6L�� ;5 �HV�XQ�FRQMXQWR�FRQWDEOH��GLVFUHWR��� HQWRQFHV� OD� YDULDEOH� DOHDWRULD� ; � VH� GHQRPLQD� YDULDEOH� DOHDWRULD�GLVFUHWD��(O�UDQJR�GH�XQD�YDULDEOH�DOHDWRULD�GLVFUHWD�SXHGH�VHU�XQ�FRQMXQWR�ILQLWR�R LQILQLWR�FRQWDEOH�FRPR�VH�LQGLFD�D�FRQWLQXDFLyQ�

Page 3: Variables aleatorias reales

L� �

LL� �

6L� � HV� HO� HVSDFLR�PXHVWUDO� GH� UHIHUHQFLD� VREUH� HO� FXDO� VH� GHILQH� OD�

YDULDEOH�DOHDWRULD� ; �\�� �HV�HO�HVSDFLR�GH�SUREDELOLGDG�LQGXFLGR�SRU�OD�

YDULDEOH�DOHDWRULD� ; �GRQGH� �HV�HO� iOJHEUD�GH�%RUHO��HQWRQFHV�VH�WLHQH�TXH�

6HD� ; XQD�YDULDEOH�DOHDWRULD�GHILQLGD�VREUH�HO�HVSDFLR�GH�SUREDELOLGDG�\�FRQ�YDORUHV�HQ�HO�HVSDFLR�PHGLEOH�� ��6L�VH�GHILQH�

�FRQ�

(QWRQFHV

�SDUD�WRGR�

6H�SXHGH�YHULILFDU�TXH� ;3 �VDWLVIDFH�ODV�FRQGLFLRQHV�TXH�GHILQHQ�XQD�PHGLGD�GH�SUREDELOLGDG�

�������

6HD�; �XQD�YDULDEOH�DOHDWRULD�GHILQLGD�VREUH�HO�HVSDFLR�GH�SUREDELOLGDG�\�VHD� ;5 �VX�UDQJR��UHFRUULGR���SDUD�FDGD� [ 5 ��VHD

(QWRQFHV�D�OD�IXQFLyQ� I VH�OD�GHQRPLQD�OD�IXQFLyQ�GH�SUREDELOLGDG� �GH�OD�YDULDEOH�DOHDWRULD� ; ��

$O�FRQMXQWR� �VH�OH�OODPD�OD�JUiILFD�GH�OD�IXQFLyQ�GH�SUREDELOLGDG�GH� ; ��OD�FXDO�VH�SXHGH�UHSUHVHQWDU�HQ�XQ�SODQR�FDUWHVLDQR�R�PHGLDQWH�XQD�WDEOD�GH�YDORUHV�WDO�FRPR�VH�LQGLFD�HQ�OD�7DEOD�����

7DEOD������)XQFLyQ�GH�SUREDELOLGDG

; �[ �[ Q[

� � � �I [ 3 ; [ �� �I [ �� �I [ � �QI [

Page 4: Variables aleatorias reales

/D�IXQFLyQ�GH�SUREDELOLGDG� I �FXPSOH�ODV�GRV�FRQGLFLRQHV�VLJXLHQWHV�

L� � � �I [ �

LL�

������

6HD� ; �XQD�YDULDEOH�DOHDWRULD�UHDO��/D�IXQFLyQ� ;) GHILQLGD�VREUH�HO�FRQMXQWR�GH�ORV�Q~PHURV�UHDOHV�5 �SRU�PHGLR�GH

6H� OODPD�OD�IXQFLyQ�GH�GLVWULEXFLyQ�GH� OD�YDULDEOH�DOHDWRULD� ; �R�GLVWULEXFLyQ�DFXPXODWLYD�GH� ; ��VL�QR�KD\�OXJDU�D�FRQIXVLRQHV�OD�IXQFLyQ�GH�GLVWULEXFLyQ�GH�OD�YDULDEOH�DOHDWRULD� ; �VH�GHQRWD�VLPSOHPHQWH�FRQ�) �

(MHPSOR����� 6H�ODQ]D�XQD�PRQHGD�WUHV�YHFHV�\�VH�REVHUYD�HO�Q~PHUR�GH�FDUDV�HQ�FDGD�XQR�GH�ORV�SRVLEOHV�UHVXOWDGRV� 'HWHUPLQDU�OD�IXQFLyQ�GH�SUREDELOLGDG�

\�OD�GLVWULEXFLyQ�GH�SUREDELOLGDG�

3DUD�REWHQHU�HO�HVSDFLR�PXHVWUDO�DVRFLDGR�D�XQ�H[SHULPHQWR�DOHDWRULR�VH�SXHGH�XWLOL]DU�XQ�GLDJUDPD�GH�iUERO��FRPR�VH�PXHVWUD�HQ�OD�)LJXUD������(Q�HVWH�HMHPSOR��HO�HVSDFLR�PXHVWUDO�HVWi�FRQIRUPDGR�SRU���UHVXOWDGRV�SRVLEOHV��VHJ~Q�VH�LQGLFD�D�FRQWLQXDFLyQ�

)LJXUD������'LDJUDPD�GH�iUERO

I �[L � ��L ;

Page 5: Variables aleatorias reales

6L�VH�GHILQH�OD�YDULDEOH�DOHDWRULD� ; �FRPR�HO�Q~PHUR�GH�FDUDV�HQ�FDGD�XQR�GH�ORV�SRVLEOHV�UHVXOWDGRV�GHO�HVSDFLR�PXHVWUDO��HQWRQFHV�

; ��1~PHUR�GH�FDUDV�HQ�FDGD�XQR�GH�ORV�SRVLEOHV�UHVXOWDGRV

� � �; FFF

� � � � � � �; FFV ; FVF ; VFF

� � � � � � �; VVF ; VFV ; FVV

� � �; VVV

$Vt� ; �HV�XQD�YDULDEOH�DOHDWRULD�GLVFUHWD�SRUTXH�VROR�WRPD�ORV�YDORUHV�HQWHURV�������������(V�GHFLU��

(O�UDQJR�GH�OD�YDULDEOH�HV�

$KRUD�

�� � ��� � ��

�I [ I 3 ;

�� � ��� � ��

�I [ I 3 ;

�� � ��� � ��

�I [ I 3 ;

�� � ��� � ��

�I [ I 3 ;

&DGD� � � �LI [

� � � ��

� � � � � � � � � �LL

I [ I [ I [ I [ I [� � � � �

�� � � � �

/D�IXQFLyQ�GH�SUREDELOLGDG�HVWi�GDGD�HQ�OD�7DEOD������VX�UHSUHVHQWDFLyQ�JUiILFD�VH�LQGLFD�HQ�OD�)LJXUD�����

� � �

[� � � �

[ [ [

Page 6: Variables aleatorias reales

7DEOD������)XQFLyQ�GH�SUREDELOLGDG�GH�OD�YDULDEOH�;

; � � � �

� � � �I [ 3 ; [ ��� ��� ��� ���

/D�GLVWULEXFLyQ�GH�SUREDELOLGDG�VH�SUHVHQWD�HQ�OD�7DEOD�����\�VX�UHSUHVHQWDFLyQ�JUiILFD�VH�LQGLFD�HQ�OD�)LJXUD������

; � � � �

� � � �) [ 3 ; [ ��� ��� ��� �

&RQ�EDVH�HQ�ODV�DQWHULRUHV�WDEODV��VH�SXHGHQ�FDOFXODU�ODV�VLJXLHQWHV�SUREDELOLGDGHV�

7DEOD������'LVWULEXFLyQ�GH�OD�YDULDEOH�DOHDWRULD�;

Page 7: Variables aleatorias reales

R

�����9DULDEOH�DOHDWRULD�FRQWLQXD

(Q�HVWH�DSDUWDGR�VH�GHILQHQ�ODV�YDULDEOHV�DOHDWRULDV�FRQWLQXDV�\�VH�HVSHFLILFDQ�VX�IXQFLyQ�GH�GHQVLGDG�GH�SUREDELOLGDG�\�VX�IXQFLyQ�GH�GLVWULEXFLyQ��6HD� ; �XQD�YDULDEOH�DOHDWRULD�GHILQLGD�VREUH�XQ�HVSDFLR�PXHVWUDO�� �\� ;5 �VX�UDQJR�HQ� 5��6L�� ;5 �HV�XQ�FRQMXQWR�QR�FRQWDEOH��LQILQLWR�QR�FRQWDEOH���HQWRQFHV�OD�YDULDEOH�DOHDWRULD� ; �SXHGH�VHU�XQD�YDULDEOH�DOHDWRULD�FRQWLQXD��,QWXLWLYDPHQWH�HO�UDQJR�GH�XQD�YDULDEOH�DOHDWRULD�FRQWLQXD�FRUUHVSRQGH�D�XQ�LQWHUYDOR�

������

8QD�IXQFLyQ�UHDO� I �TXH�VDWLVIDFH�ODV�GRV�FRQGLFLRQHV�VLJXLHQWHV�

L� � � �I [

LL� � � �I [ G[

VH�GHQRPLQD�XQD�IXQFLyQ�GH�GHQVLGDG�GH�SUREDELOLGDG� SDUD�OD�YDULDEOH�DOHDWRULD�FRQWLQXD� ; ��

6HD� ; �XQD�YDULDEOH�DOHDWRULD�WDO�TXH�SDUD�WRGR�HYHQWR� �VH�FXPSOH�TXH�

� � � �;

%

3 % I [ G[

3DUD�DOJXQD�IXQFLyQ�GH�GHQVLGDG� I ��HQWRQFHV�OD�YDULDEOH�DOHDWRULD� ; �VH�OODPD�XQD�YDULDEOH�DOHDWRULD�GH�WLSR�FRQWLQXD�R�XQD�YDULDEOH�DOHDWRULD�FRQWLQXD��(V�GH�VHxDODU�TXH�% �HV�XQ�FRQMXQWR�%RUHOLDQR�\�OD�LQWHJUDO�VREUH�% HV�XQD�LQWHJUDO�GH�/HEHVJXH��6KDR��������

Page 8: Variables aleatorias reales

�������

6HD� ; � XQD� YDULDEOH� DOHDWRULD� UHDO� GHILQLGD� VREUH� HO� HVSDFLR� GH� SUREDELOLGDG���6H�GLFH�TXH� ; � HV�DEVROXWDPHQWH�FRQWLQXD�� VL�\� VROR� VL�� H[LVWH�XQD�

IXQFLyQ�UHDO�QR�QHJDWLYD�H�LQWHJUDEOH� ;I ��WDO�TXH�SDUD�WRGR� [ 5 ��VH�VDWLVIDFH�

/D�IXQFLyQ� ;I �VH�GHQRPLQD�IXQFLyQ�GH�GHQVLGDG�GH�SUREDELOLGDG� GH�

OD�YDULDEOH�DOHDWRULD� ; ��$�OD�IXQFLyQ� ;) �VH�OH�OODPD�IXQFLyQ�GH�GLVWULEXFLyQ�GH�

OD�YDULDEOH�DOHDWRULD� ; R�IXQFLyQ�GH�GLVWULEXFLyQ�DFXPXODWLYD�GH�SUREDELOLGDG�

(MHPSOR�������$QDOL]DU�VL�OD�IXQFLyQ�GDGD�SRU�

D� (V� XQD� IXQFLyQ� GH� GHQVLGDG� SDUD� OD� YDULDEOH� DOHDWRULD� ; �� E�� GH� VHU� DVt�GHWHUPLQDU�OD�IXQFLyQ�GH�GLVWULEXFLyQ�SDUD� ; ��

6ROXFLyQ�D��SDUD�YDORUHV�GH� [ �IXHUD�GHO�LQWHUYDOR�>���@�OD�IXQFLyQ�YDOH�FHUR�\�

SDUD�YDORUHV�GH� [ �HQ�HO�LQWHUYDOR�>���@�VH�WUDEDMD�FRQ��

�� � �I [ [

(Q�OD�7DEOD�����VH�SUHVHQWDQ�DOJXQRV�YDORUHV�GH�OD�IXQFLyQ� � �I [ �\�HQ�OD�)LJXUD�

����VH�PXHVWUD�OD�UHSUHVHQWDFLyQ�JUiILFD�GH�OD�PHQFLRQDGD�IXQFLyQ�

7DEOD������)XQFLyQ�GH�GHQVLGDG�GH�SUREDELOLGDG�SDUD�OD�YDULDEOH�DOHDWRULD�;

[ � ��� ��� � �� �� � ���

� �I [ � ���� ���� � � � � �

Page 9: Variables aleatorias reales

)LJXUD������)XQFLyQ�GH�GHQVLGDG�GH�SUREDELOLGDG�SDUD�OD�YDULDEOH�DOHDWRULD ;

'H�OD�GHILQLFLyQ�GH�OD�IXQFLyQ�\�GH�OD�)LJXUD������VH�YH�TXH� � � �I [ �

$GHPiV�� �

� �

� � � � � � � �I [ G[ I [ G[ I [ G[ I [ G[

� ��

� �

� � � � �I [ G[ G[ [ G[ G[�

��

�_[

�� � �

�� � � � �_I [ G[ [

$Vt��

� � �I [ G[

(Q�FRQFOXVLyQ�� � �I [ �Vt�HV�XQD�IXQFLyQ�GH�GHQVLGDG�SDUD�OD�YDULDEOH�DOHDWRULD� ; �

E� /D�IXQFLyQ�GH�GLVWULEXFLyQ�SDUD� ; �HV�OD�VLJXLHQWH�

��������� � �

� � ������ �� �

���������VL� �;

VL [

) [ [ VL [

[

Page 10: Variables aleatorias reales

/D�UHSUHVHQWDFLyQ�JUiILFD�GH�OD�IXQFLyQ�GH�GLVWULEXFLyQ�SDUD�OD�YDULDEOH�DOHDWRULD�;�VH�LQGLFD�HQ�OD�)LJXUD�����

6HD� ; XQD� YDULDEOH� DOHDWRULD� UHDO� GHILQLGD� VREUH� HO� HVSDFLR� GH� SUREDELOLGDG���OD�IXQFLyQ�GH�GLVWULEXFLyQ� ;) �VDWLVIDFH�ODV�VLJXLHQWHV�FRQGLFLRQHV�

L� � � � �;) [ �SDUD�WRGR� [ 5 �

LL� 6L � �[ [ �HQWRQFHV� � �� � � �; ;) [ ) [ �

LLL��

� � OLP � � � �; ; ;D

) [ ) [ D ) [ �

LY� OLP � � �;[

) [ �

Y� OLP � � �;[

) [ �

�����(VSHUDQ]D� DWHPiWLFD�\�YDULDQ]D�GH�XQD�YDULDEOH�DOHDWRULD

/RV�VLJXLHQWHV�FRQFHSWRV�UHIHUHQWHV�D�OD�HVSHUDQ]D�PDWHPiWLFD�\�OD�YDULDQ]D�GH�XQD�YDULDEOH�DOHDWRULD�VRQ�DGRSWDGRV�FRQ�EDVH�HQ�OR�H[SXHVWR�/LQGJUHQ���������6KDR��������\�%XUEDQR��������

6HD� ; � XQD� YDULDEOH� DOHDWRULD� UHDO� GHILQLGD� VREUH� HO� HVSDFLR� GH� SUREDELOLGDG�

��HQWRQFHV�

)LJXUD������)XQFLyQ�GH�GLVWULEXFLyQ�SDUD�OD�YDULDEOH�DOHDWRULD�;�

Page 11: Variables aleatorias reales

L� 6L� ; HV�XQD�YDULDEOH�DOHDWRULD�GLVFUHWD�FRQ� UDQJR \� I � VX

IXQFLyQ�GH�SUREDELOLGDG�HQWRQFHV��HO�YDORU�HVSHUDGR�GH� ; �HVWi�GDGR�SRU�

6LHPSUH�\�FXDQGR�OD�DQWHULRU�VXPD�H[LVWD�

LL� 6L�; �HV�XQD�YDULDEOH�DOHDWRULD�FRQWLQXD�FRQ�IXQFLyQ�GH�GHQVLGDG� ;I �HQWRQFHVHO�YDORU�HVSHUDGR�GH� ; �R�PHGLD�GH�OD�YDULDEOH�DOHWRULD�HVWi�GDGR�SRU��

� � � �;( ; [I [ G[

6LHPSUH�\�FXDQGR�OD�DQWHULRU�LQWHJUDO�H[LVWD�

�������

6L� ; � HV� XQD� YDULDEOH� DOHDWRULD� GHILQLGD� VREUH� HO� HVSDFLR� GH� SUREDELOLGDG��

��DOJXQDV�SURSLHGDGHV�GH�OD�HVSHUDQ]D�PDWHPiWLFD�VRQ�

L� 3DUD�FXDOTXLHU� D 5 ��HQWRQFHV� � �( D D �

LL� 3DUD�FXDOTXLHU�D 5 ��HQWRQFHV� � � � �( D; D( ; �

LLL� � � � � � �( D; E< D( ; E( < �

&RPR�FRQVHFXHQFLD��GH�OD�SURSLHGDG�LLL���VH�WLHQH�

� � � � � �( ; < ( ; ( <

6LHQGR� �; < YDULDEOHV� DOHDWRULDV� GHILQLGDV� VREUH� HO� PLVPR� HVSDFLR� GH�SUREDELOLGDG�

6L� ; �HV�XQD�YDULDEOH�DOHDWRULD�UHDO�GHILQLGD�VREUH�HO�HVSDFLR�GH�SUREDELOLGDG���HQWRQFHV�OD�YDULDQ]D�GH� ; �VH�GHQRWD�\��GHILQH�SRU�

6LHPSUH�TXH�HO�YDORU�HVSHUDGR�GH� ; �H[LVWD��$O�Q~PHUR�

Page 12: Variables aleatorias reales

VH�OH�OODPD�OD�GHVYLDFLyQ�HVWiQGDU�GH�OD�YDULDEOH�DOHDWRULD ; �

�������

(QWUH�ODV�SURSLHGDGHV�GH�OD�YDULDQ]D�GH�OD�YDULDEOH�DOHDWRULD� ;� HVWiQ��

L� 3DUD�FXDOTXLHU�D 5 ��HQWRQFHV� � � �9DU D �

LL� 3DUD�FXDOTXLHU�D 5 ��HQWRQFHV� �� � � �9DU D; D 9DU ; �

LLL� � � �9DU ; �

LY� 3DUD�FXDOTXLHU�D 5 ��HQWRQFHV� � � � �9DU D ; 9DU ; �

Y� � � �� �VL�\�VyOR�VL�� � � �� �9DU ; 3 ; ( ;

YL� 6L� ; �HV�XQD�YDULDEOH�DOHDWRULD�WDO�TXH� � �( ; �H[LVWH��HQWRQFHV

(MHPSOR���� �3DUD�OD�YDULDEOH�DOHDWRULD�GLVFUHWD�DVRFLDGD�DO�H[SHULPHQWR�DOHDWRULR�GH�ODQ]DU�XQD�PRQHGD�WUHV�YHFHV�\�GH�REVHUYDU�HO�Q~PHUR�GH�FDUDV�HQ�FDGD�XQR�GH� ORV�SRVLEOHV� UHVXOWDGRV��FRPR�VH� LQGLFy�HQ�HO�HMHPSOR������� OD� � IXQFLyQ�GH�SUREDELOLGDG�VH�LQGLFD�HQ�OD�7DEOD�����

7DEOD������)XQFLyQ�GH�SUREDELOLGDG

; � � � �

� � � �I [ 3 ; [ ��� ��� ��� ���

&RPR�HO�YDORU�HVSHUDGR�GH�OD�YDULDEOH�DOHDWRULD� ; �HVWi�GDGR�SRU�

� � � � � �L ; L ;

L L L L[ 5 [ 5

( ; [ I [ [ 3 ; [

HQWRQFHV�

3DUD�GHWHUPLQDU�OD�YDULDQ]D��LQLFLDOPHQWH�VH�FDOFXOD�

Page 13: Variables aleatorias reales

� � �� � � � � �L ; L ;

L L L L[ 5 [ 5

( ; [ I [ [ 3 ; [

� � � � �� � � � ��� � � � � � �

� � � � �( ;

&RPR

�� �

� � �� �

9DU ;

/D�GHVYLDFLyQ�HVWiQGDU�HV�

(MHPSOR������3DUD�OD�YDULDEOH�DOHDWRULD�FRQWLQXD�GHO�HMHPSOR������VH�WLHQH�TXH�OD�IXQFLyQ�H[SUHVDGD�SRU�

HV�XQD�IXQFLyQ�GH�GHQVLGDG�SDUD� OD�YDULDEOH�DOHDWRULD� ; ��&DOFXODU�VX� � �( ; �� �9DU ; �\�GHVYLDFLyQ�HVWiQGDU��

(O�YDORU�HVSHUDGR�GH�OD�YDULDEOH�DOHDWRULD� ; �R�PHGLD�GH�OD�YDULDEOH�DOHDWRULD�HV�

� � � �( ; [I [ G[

(QWRQFHV�� �

� �

� � � � � � � �( ; [I [ G[ [I [ G[ [I [ G[

� ��

� �

� � ��� �� � ���( ; [ G[ [ [ G[ [ G[

� ���

��

� � � ��_[

( ; [ G[

Page 14: Variables aleatorias reales

� � ��

� � �� � � � � ����

� � � �_[

( ;

� �� � � �( ; [ I [ G[

� �� � � � �

� �

� � ��� �� � ���( ; [ G[ [ [ G[ [ G[

� � � ��� �

��

� � �� � � � � � ���

� � � �_[

( ; [ G[

/D�YDULDQ]D�GH� ; �HVWi�GDGD�SRU�

� � �� � � � > � �@ ��� ���� ��� ������ ������9DU ; ( ; ( ;

'HVYLDFLyQ�HVWiQGDU�GH� ; �HV�

&XDQGR� ; �HV�XQD�YDULDEOH�DOHDWRULD�FRQWLQXD���HOOD�WRPD�FXDOTXLHU�YDORU�HQ�XQ�LQWHUYDOR�GH�Q~PHURV�UHDOHV��HV�GHFLU��¦VX�UDQJR�HV�XQ�VXEFRQMXQWR�QR�QXPHUDEOH��GH� Q~PHURV� UHDOHV§�� (Q� OD� SUiFWLFD� DOJXQRV� HMHPSORV� GH� YDULDEOHV� DOHDWRULDV�UHDOHV�SRGUtDQ�VHU�ORV�VLJXLHQWHV�

D� ; ��OD�GXUDFLyQ�GH�XQ�ERPELOOR�HOHJLGR�DO�D]DU�GH�XQ�GHWHUPLQDGR�ORWH�� ;SXHGH�WRPDU�YDORUHV�HQ�>��� ��

E� ; ��OD�HVWDWXUD�HQ�PHWURV�GH�XQD�SHUVRQD�HVFRJLGD�DO�D]DU�HQ�HO���FROHJLR���$�; �SXHGH���WRPDU���YDORUHV��HQ��HO�LQWHUYDOR�>��������@�

F� ; ��HO�SHVR�HQ�JUDPRV�GH�XQ�REMHWR�WRPDGR�DO�D]DU�� ; �SRGUtD�WRPDU�YDORUHVHQ�HO�LQWHUYDOR�>�������@�

G� ; ��ORQJLWXG�HQ�FHQWtPHWURV�GH�XQ�WDOOR�HOHJLGR�DO�D]DU�HQ�XQ�MDUGtQ��; �SRGUtDWRPDU�YDORUHV�HQ�>������@�

(V�FRQYHQLHQWH�VHxDODU�TXH�VL� � �I [ �HV�XQD�IXQFLyQ�GH�GHQVLGDG�SDUD�OD�YDULDEOH�DOHDWRULD� ; �� HQWRQFHV� OD� SUREDELOLGDG� GH� TXH� OD� YDULDEOH� DOHDWRULD� ; � WRPH�YDORUHV�HQ�XQ�LQWHUYDOR� HVWi�GDGD�SRU�

� � � � � � � � � � ��E

D

3 D ; E 3 D ; E 3 D ; E 3 D ; E I [ G[

Page 15: Variables aleatorias reales

(MHPSOR������3DUD�OD�YDULDEOH�DOHDWRULD�FRQWLQXD�FRQVLGHUDGD�HQ�HO�HMHPSOR�������VH�TXLHUH�FDOFXODU�ODV�VLJXLHQWHV�SUREDELOLGDGHV�� �� ����3 ; �� � � ����3 ; �

(Q�HIHFWR���� ���

���� � � �

�� �

�� ���� � � � ����� ��� �����_3 ; I [ G[ [ G[ [

� ��� � ������� �

�� � � �

� � ���� � � � � � � _3 ; I [ G[ I [ G[ G[ [ G[ [

� �� � ���� ����� ��� �����3 ; �

�����0RPHQWRV�GH�XQD�YDULDEOH�DOHDWRULD

6HD� ; �XQD�YDULDEOH�DOHDWRULD��UHDO��(O�U�pVLPR�PRPHQWR�FHQWUDO�GH ; �DOUHGHGRU�GH�FHUR��VH�GHQRWD�\�GHILQH�DVt�

6LHPSUH�\�FXDQGR�HO�YDORU�HVSHUDGR�H[LVWD�

6HD� ; � XQD� YDULDEOH� DOHDWRULD� � UHDO� FX\R� YDORU� HVSHUDGR� H[LVWH�� (O� U�pVLPR�PRPHQWR�FHQWUDO�GH ; �DOUHGHGRU�GH� � �( ; ��VH�GHQRWD�\�GHILQH�DVt�

6LHPSUH�\�FXDQGR�HO�YDORU�HVSHUDGR�H[LVWD�

(Q�FRQFRUGDQFLD� FRQ�%XUEDQR�HW� DO�� �������� SDUD� HO� FDVR�GLVFUHWR� HO� U�pVLPR�PRPHQWR�FHQWUDO�GH ; �DOUHGHGRU�GH�VX�YDORU�HVSHUDGR� � �( ; �VH�SXHGH�H[SUHVDU�GH�OD�VLJXLHQWH�PDQHUD�

$GHPiV��SDUD�HO�FDVR�FRQWLQXR�HV�

'H� DFXHUGR� FRQ� $EUDPRZLW]� \� 6WHJXQ� �������� HO� FRHILFLHQWH� GH� DVLPHWUtD�\�HO�FRHILFLHQWH�GH�FXUWRVLV� VH�GHILQHQ�D� WUDYpV�GH� ODV� VLJXLHQWHV�H[SUHVLRQHV��UHVSHFWLYDPHQWH��

Page 16: Variables aleatorias reales

(MHPSOR�����3DUD�OD�YDULDEOH�DOHDWRULD ; GLVFUHWD�FRQVLGHUDGD�HQ�HO�HMHPSOR������HO�FRHILFLHQWH�GH�DVLPHWUtD�VH�REWLHQH�UHDOL]DQGR�HO�VLJXLHQWH�SURFHGLPLHQWR��

&RPR�

(O�UHVXOWDGR�DQWHULRU�LQGLFD�TXH�HO�FRHILFLHQWH�GH�DVLPHWUtD�GH�OD�YDULDEOH�DOHDWRULD�;�HV�LJXDO�D�FHUR�

(MHPSOR�����3DUD�OD�YDULDEOH�DOHDWRULD ; GLVFUHWD�FRQVLGHUDGD�HQ�HO�HMHPSOR������HO�FRHILFLHQWH�GH�FXUWRVLV�VH�REWLHQH�PHGLDQWH�HO�VLJXLHQWH�FRQMXQWR�GH�SDVRV�

Page 17: Variables aleatorias reales

&RPR��

(O�UHVXOWDGR�DQWHULRU�LQGLFD�TXH�HO�FRHILFLHQWH�GH�FXUWRVLV�GH�OD�YDULDEOH�DOHDWRULD�;�HV�LJXDO�D��������DSUR[LPDGDPHQWH�

(MHPSOR������(Q�HO�VLJXLHQWH�FDVR�VH�UHTXLHUH��D��GHWHUPLQDU�VL�OD�IXQFLyQ�GDGD�FRUUHVSRQGH�D�XQD�IXQFLyQ�GH�GHQVLGDG�GH�SUREDELOLGDG�SDUD�OD�YDULDEOH�DOHDWRULD�;��E��GH�VHU�DVt�FDOFXODU�VX�YDORU�HVSHUDGR�\�VX�YDULDQ]D��F��GHWHUPLQDU�VX�IXQFLyQ�GH�GLVWULEXFLyQ�GH�SUREDELOLGDG��G��GHWHUPLQDU�ODV�SUREDELOLGDGHV�3�;������\�3�;�!������

����� � �� �

���������HQ�RWUR�FDVR

[H VL [I [

D� (Q� OD�7DEOD� ���� VH� SUHVHQWDQ� DOJXQRV� YDORUHV� GH� OD� IXQFLyQ� � �I [ � \� HQ� OD)LJXUD�����VH�PXHVWUD�OD�UHSUHVHQWDFLyQ�JUiILFD�GH�OD�PHQFLRQDGD�IXQFLyQ��

I�[�

[ �� �� � � � � �

� �I [ � � � ����� ������ ����� �����

)LJXUD������)XQFLyQ�GH�GHQVLGDG�GH�SUREDELOLGDG�SDUD�OD�YDULDEOH�DOHDWRULD�;

Page 18: Variables aleatorias reales

'H�OD�GHILQLFLyQ�GH�OD�IXQFLyQ�\�GH�OD�)LJXUD������VH�YH�TXH� � � �I [ �$GLFLRQDOPHQWH�

/XHJR��

� � �I [ G[ �

(Q�FRQFOXVLyQ�� � �I [ �VL�HV�XQD�IXQFLyQ�GH�GHQVLGDG�SDUD�OD�YDULDEOH�DOHDWRULD� ; �

E� /D�IXQFLyQ�GH�GLVWULEXFLyQ�SDUD� ; �HV�OD�VLJXLHQWH�

��������� � �� �

� ������ � �; [

VL [) [

H VL [

/D�UHSUHVHQWDFLyQ�JUiILFD�GH�OD�IXQFLyQ�GH�GLVWULEXFLyQ�SDUD�OD�YDULDEOH�DOHDWRULD�;�VH�LQGLFD�HQ�OD�)LJXUD����

)LJXUD������)XQFLyQ�GH�GHQVLGDG�GH�SUREDELOLGDG�SDUD�OD�YDULDEOH�DOHDWRULD�;

F� (O�YDORU�HVSHUDGR�GH�OD�YDULDEOH�DOHDWRULD� ; HV�

� � � �( ; [I [ G[

Page 19: Variables aleatorias reales

(QWRQFHV� �

� � � � � �( ; [I [ G[ [I [ G[

� � ��� [( ; [ G[ [H G[

$KRUD�LQWHJUDQGR�SRU�SDUWHV��UHVXOWD�

/XHJR�

$GLFLRQDOPHQWH�� �� � � �( ; [ I [ G[

�� � �

� � ��� [( ; [ G[ [ H G[

�� �� � [( ; [ H G[

,QWHJUDQGR�GRV�YHFHV�SRU�SDUWHV��VH�WLHQH�

/D�YDULDQ]D�GH� ; �HVWi�GDGD�SRU�

� � �� � � � > � �@ � ��� � � �9DU ; ( ; ( ;

'HVYLDFLyQ�HVWiQGDU�GH� ; �HV�

G� $�FRQWLQXDFLyQ�VH�FDOFXODUiQ�ODV�SUREDELOLGDGHV�VLJXLHQWHV��3�;������\�3�;!������

Page 20: Variables aleatorias reales

(Q�FRQVHFXHQFLD� ���� ���� ������3 ; H

� �

� �� � � � �3 ; I [ G[ I [ G[

� �

� �� ��� [3 ; G[ H G[

3RU�OR�WDQWR��� �� � ������3 ; H

�����)XQFLyQ�JHQHUDGRUD�GH�PRPHQWRV

6HD� ; � XQD� YDULDEOH� DOHDWRULD� WDO� TXH� � �W;( H � HV� ILQLWR� SDUD� WRGR���6H�GHILQH� OD� IXQFLyQ�JHQHUDGRUD�GH�PRPHQWRV�GH� ;

DVt��%ODQFR��������

6L� ; �HV�XQD�YDULDEOH�DOHDWRULD�GLVFUHWD�WDO�TXH

L�

(QWRQFHV�

� � � �L

QW[

; LL

P W H 3 ; [

6L� ; �HV�XQD�YDULDEOH�DOHDWRULD�GLVFUHWD�WDO�TXH

LL�

(QWRQFHV�

� � � �LW[; L

L

P W H 3 ; [

Page 21: Variables aleatorias reales

6L� ; �HV�XQD�YDULDEOH�DOHDWRULD�FRQWLQXD�FRQ�IXQFLyQ�GH�GHQVLGDG� I ��HQWRQFHV

� � � �W[;P W H I [ G[

/D�H[LVWHQFLD�GH�OD�IXQFLyQ�JHQHUDGRUD�GH�PRPHQWRV�GH�XQD�YDULDEOH�DOHDWRULD�;JDUDQWL]D�OD�H[LVWHQFLD�GH�WRGRV�ORV�PRPHQWRV�GH�RUGHQ�U�DOUHGHGRU�GH�FHUR�GH�OD�YDULDEOH� ; ��$GHPiV�VL�OD�IXQFLyQ�JHQHUDGRUD�GH�PRPHQWRV�H[LVWH��HQWRQFHV�HV�GLIHUHQFLDEOH�HQ�XQD�YHFLQGDG�GHO�RULJHQ��HQ�HVWH�FDVR�GH�FHUR��\�VH�FXPSOH�TXH�

�� � � �U

U; WU

GP W ( ;

GW

6L� ; �HV�XQD�YDULDEOH�DOHDWRULD�FX\D�IXQFLyQ�JHQHUDGRUD�GH�PRPHQWRV� ���;PH[LVWH��HQWRQFHV��H[LVWH� D 5 WDO�TXH�

� � � � ��SDUD�WRGR� � � ��

LL

;L

WP W ( ; W D D

L

$GHPiV��

�� � � �U

U; WU

GP W ( ;

GW

6L� ; �� < VRQ� YDULDEOHV� DOHDWRULDV� FX\DV� IXQFLRQHV� JHQHUDGRUDV� GH�PRPHQWRV����;P �� ���<P �H[LVWHQ�\�DGHPiV�

� � � �� ��SDUD�WRGR�� �; <P W P W W

HQWRQFHV� ;� < �WLHQHQ�OD�PLVPD�GLVWULEXFLyQ�

(MHPSOR� �����6HD� ; � XQD� YDULDEOH� DOHDWRULD� FX\R� UDQJR� HV� �� FRQ�IXQFLyQ�GH�SUREDELOLGDG�GDGD�SRU�

�������� � �� �

� ���VL� �;

S VL [I [

S [

3DUD�HVWH�FDVR��VH�WLHQH�TXH� ��� � ��;I 3 ; S �\� ��� � �� �;I 3 ; SFRUUHVSRQGHQ�D�ODV�SUREDELOLGDGHV�GH�p[LWR�\�GH�IUDFDVR�HQ�HO�PRGHOR�GH�%HUQRXOOL���VH�WUDWDUi�HQ�XQ�FDStWXOR�SRVWHULRU��

/D�IXQFLyQ�JHQHUDGRUD�GH�PRPHQWRV�GH� ; �HV���

� � � �LW[; L

L

P W H 3 ; [

Page 22: Variables aleatorias reales

��� ���� � � �� � ��W W;P W H 3 ; H 3 ;

� � � W;P W S SH

(O�YDORU�HVSHUDGR�VH�SXHGH�REWHQHU�GH�OD�VLJXLHQWH�PDQHUD�

SRU�OR�WDQWR� �( ; S

7DPELpQ�VH�WLHQH�TXH�

&RPR�� � �� � � � > � �@ �� �9DU ; ( ; ( ; S S S S

HQWRQFHV�� � �� �9DU ; S S

(MHPSOR������6HD� ; �XQD�YDULDEOH�DOHDWRULD�FX\R�UDQJR�HV� ��Q = FX\D�IXQFLyQ�GH�SUREDELOLGDG�HV�

���� � �������� � ��������

�� ��� �

� HQ�RWUR�FDVR

[ Q [

;

QS S VL [ Q

[ Q [I [

3DUD�HVWH�FDVR��VH�WLHQH�TXH�� S �\� � S �FRUUHVSRQGHQ�D�ODV�SUREDELOLGDGHV�GH�p[LWR�\�GH� IUDFDVR�HQ�HO�PRGHOR�GH�%LQRPLDO�HO�FXDO� VH� WUDWDUi�HQ�HO�FDStWXOR�SRVWHULRU�

/D�IXQFLyQ�JHQHUDGRUD�GH�PRPHQWRV�GH� ; �HV�

� �

� �

� � � � � �L

Q QW[ W L

; LL L

P W H 3 ; [ H 3 ; L

� � � �

� �

�� � ��� � ��� �

�� ��

Q QW L L Q L W L L Q L

;L L

QQP W H S S H S S

LL Q L

Page 23: Variables aleatorias reales

(O�YDORU�HVSHUDGR�VH�SXHGH�REWHQHU�GH�OD�VLJXLHQWH�IRUPD�

3RU�OR�WDQWR� �( ; QS

7DPELpQ�VH�WLHQH�TXH�

&RPR� � � � � �� � � � > � �@ � � �� �9DU ; ( ; ( ; Q S QS QS QS QS S

HQWRQFHV�

� � �� �9DU ; QS S

(MHPSOR������8QD�YDULDEOH�DOHDWRULD� ; WLHQH�FRPR�IXQFLyQ�GH�GHQVLGDG��

����� � �� �

���������HQ�RWUR�FDVR

[H VL [I [

/D�IXQFLyQ�JHQHUDGRUD�GH�PRPHQWRV�HV

� � � � ��� � �W[ W[ W[;P W H I [ G[ H G[ H I [ G[

�� � �� ��

� �

�� �

�W[ [ [ W [ W

;P W H H G[ H G[ HW

Page 24: Variables aleatorias reales

�� � �� �� �

� � �� �

� � �[ W [ W E

;E

P W H /LP HW W W

�� � ����VL� ���

�;P W WW

(O�YDORU�HVSHUDGR�VH�SXHGH�REWHQHU�GH�OD�VLJXLHQWH�IRUPD�

� � ��

� �� � � � �

� �� �; W W W

G G( ; P W

GW GW W W3RU�OR�WDQWR

� � �( ;

7DPELpQ�VH�WLHQH�TXH��� �

�� � �� � � �

� � �� � � � ���

� �� � �� �; W W W

G G G( ; P W

GW GW W GW W W

&RPR�

HQWRQFHV�

� � �9DU ;

(VWRV� UHVXOWDGRV� FRQFXHUGDQ� FRQ� ORV� YDORUHV� GH� OD� HVSHUDQ]D� \� OD� YDULDQ]D�GHWHUPLQDGRV�HQ�HO�HMHPSOR�����

(MHPSOR������8QD�YDULDEOH�DOHDWRULD� ; WLHQH�FRPR�IXQFLyQ�GH�GHQVLGDG��

�� �

���������HQ�RWUR�FDVR

VL D [ EI [ E D

/D�IXQFLyQ�JHQHUDGRUD�GH�PRPHQWRV�HV�

� � � � ��� � � ���D E

W[ W[ W[ W[;

D E

P W H I [ G[ H G[ H I [ G[ H G[

� �� �

� � � �

E EW DWW[ W[ E

; D

D

H HP W H G[ H

E D E D W E D W

� �� �

EW DW

;

H HP W

E D W

Page 25: Variables aleatorias reales

/D�DQWHULRU�H[SUHVLyQ�UHTXLHUH�TXH� �W ��SRU�OR�WDQWR�OD�IXQFLyQ�JHQHUDGRUD�GH�PRPHQWRV�QR�H[LVWH��

(V� QHFHVDULR� LQGLFDU� TXH� VL� ; �� < � VRQ� YDULDEOHV� DOHDWRULDV� LQGHSHQGLHQWHV� \����;P � ���<P � VXV� FRUUHVSRQGLHQWHV� IXQFLRQHV� JHQHUDGRUDV� GH� PRPHQWRV��

HQWRQFHV�� � � � � �� ��SDUD�WRGR�W�; < ; <P W P W P W

(Q��HIHFWR�� � �� � � � � � � � � � � � � �� �SDUD�WRGR�W�W ; < W; W< W; W<

; < ; <P W ( H ( H H ( H ( H P W P W

�����)XQFLyQ�FDUDFWHUtVWLFD

6HD� ; �XQD�YDULDEOH�DOHDWRULD��/D�IXQFLyQ�FDUDFWHUtVWLFD�GH� OD�YDULDEOH� ; �� VH�GHQRWD��\�VH�GHILQH�GH�OD�VLJXLHQWH�IRUPD�

�; 5 &

7DO�TXH� � � � �FRV� �� � � ����SDUD� �LW;

; W ( H ( W; L( VHQ W; L

(MHPSOR������ 6HD� ; �XQD�YDULDEOH�DOHDWRULD�FRQ�UDQJR� \�IXQFLyQ�GH�SUREDELOLGDG�GDGD�SRU� �

� ���

3 ;

�� ��

�3 ;

(QWRQFHV�OD�IXQFLyQ�FDUDFWHUtVWLFD�VH�REWLHQH�GHO�VLJXLHQWH�PRGR�

� �� � � � �FRV��� FRV� �� � ��� � ��

� �LW;

; W ( H W L VHQ VHQ W

� �� � � � �� FRV� �� � �

� �LW;

; W ( H W LVHQ W

�����&XDQWLOHV�\�PHGLDQD�GH�XQD�YDULDEOH�DOHDWRULD

6HD� � � � �) [ 3 ; [ �XQD�IXQFLyQ�GH�GLVWULEXFLyQ�GH�XQD�YDULDEOH�DOHDWRULD�� ;

\�VHD�� �VL�H[LVWH�XQ�Q~PHUR�UHDO� T[ 5 WDO�TXH��

� �T) [ T

Page 26: Variables aleatorias reales

(QWRQFHV� T[ �VH�GHQRPLQD�HO�FXDQWLO�T �GH�OD�YDULDEOH�DOHDWRULD� ; �

6H�GHILQH�OD�PHGLDQD�GH�;��R�GH�)��FRPR�HO��SXQWR� SDUD�HO�FXDO�VH�FXPSOH�TXH�

/D� LJXDOGDG� VROR� VH� FXPSOH� FXDQGR� ) � HV� XQD� IXQFLyQ� GH� GLVWULEXFLyQ� FRQ�GHQVLGDG�FRQWLQXD�\�HQ�HVWH�FDVR�OD�PHGLDQD�HV�~QLFD��

$O� FRQMXQWR� GH� ODV� IXQFLRQHV� GH� GLVWULEXFLyQ� DEVROXWDPHQWH� FRQWLQXDV� FRQ�PHGLDQD�FHUR��VH�OR�GHQRWD�SRU��+HWWPDQVSHUJHU��������

(MHPSOR������ 6HD� ; �XQD�YDULDEOH�DOHDWRULD�FRQ�UDQJR� \�IXQFLyQ�GH�SUREDELOLGDG�GDGD�SRU�

����� ���I I

/D�IXQFLyQ�GH�GLVWULEXFLyQ�GH�OD�YDULDEOH�DOHDWRULD� ; HVWi�GDGD�SRU�

�����VL� �

�� � ���VL�� �

�������VL�� �

;

[

) [ [

[

'HELGR�D�TXH�H[LVWHQ� LQILQLGDG�GH�YDORUHV� S[ T � TXH� VDWLVIDFHQ� OD� VLJXLHQWH�LJXDOGDG�

�� � � � � �

�; S S S) [ 3 ; [ 3 ; [

6H�FRQFOX\H�TXH�KD\�LQILQLGDG�GH�PHGLDQDV��SRU�HMHPSOR��

�S[ T ���

�S[ TVRQ�DOJXQDV�PHGLDQDV�GH�OD�YDULDEOH� ; ��

(MHPSOR������ 8QD�YDULDEOH�DOHDWRULD� ; �WLHQH�FRPR�IXQFLyQ�GH�GHQVLGDG�

�������� ��� �

� � ����������HQ�RWUR�FDVR

VL [I [

Page 27: Variables aleatorias reales

/D�IXQFLyQ�GH�GLVWULEXFLyQ�GH�OD�YDULDEOH�DOHDWRULD� ; HVWi�GDGD�SRU�

� VL� �

�� � � �����VL��� �

�����������������VL�� �

;

[

) [ [ [

[

(O�YDORU� �S[ T �VDWLVIDFH�OD�VLJXLHQWH�LJXDOGDG�

�� � � � � �

�; S S S) [ 3 ; [ 3 ; [

3RU�OR�WDQWR� �S[ T �HV�OD�PHGLDQD�GH�OD�YDULDEOH� ;

Page 28: Variables aleatorias reales

���� 3DUD� HO� HVSDFLR� PXHVWUDO� FRQVWLWXLGR� SRU� ORV� VLJXLHQWHV� UHVXOWDGRV��QR�� Vt�� FRPR� OD� SRVLEOH� UHVSXHVWD� TXH� SRGUtD� GDU� XQD� SHUVRQD� HVFRJLGD�DOHDWRULDPHQWH�D�OD�SUHJXQWD���HV�XVWHG�XQ�WUDEDMDGRU��GHO�VHFWRU�S~EOLFR"

6H�GHILQH�OD�VLJXLHQWH�YDULDEOH�DOHDWRULD�

���

Q

V

'HWHUPLQDU� D�� HO� UDQJR� GH� OD� YDULDEOHV� E�� OD� IXQFLyQ� GH� SUREDELOLGDG�� F�� OD�GLVWULEXFLyQ�GH�SUREDELOLGDG��G��HO�YDORU�HVSHUDGR��H��OD�YDULDQ]D��I��OD�GHVYLDFLyQ�HVWiQGDU� GH� OD� YDULDEOH� DOHDWRULD� \� J�� FDOFXODU� OD� SUREDELOLGDG�� ��

�� �

����6H�VHOHFFLRQDQ�DOHDWRULDPHQWH�GRV�DUWtFXORV�GH�XQ�SURFHVR�SURGXFWLYR�SDUD�DQDOL]DU�VL�VRQ�GHIHFWXRVRV�R�QR�GHIHFWXRVRV�\�VH�REWLHQH�HO�VLJXLHQWH�HVSDFLR�PXHVWUDO�� �� GRQGH� G �� DUWtFXOR� GHIHFWXRVR�� Q �� DUWtFXOR� QR�GHIHFWXRVR��6H�GHILQH�OD�VLJXLHQWH�YDULDEOH�DOHDWRULD��

�; 5

; ��Q~PHUR�GH�DUWtFXORV�GHIHFWXRVRV�HQ�FDGD�UHVXOWDGR�GH� �

'HWHUPLQDU�D� D�IXQFLyQ�GH�SUREDELOLGDGE� D�GLVWULEXFLyQ�GH�SUREDELOLGDGF� O�YDORU�HVSHUDGRG� D�YDULDQ]DH� D�GHVYLDFLyQ�HVWiQGDU�GH�OD�YDULDEOH�DOHDWRULDI� DOFXODU�ODV�VLJXLHQWHV�SUREDELOLGDGHV� �� �\�

�����$QDOL]DU�VL�OD�IXQFLyQ�GDGD�SRU�

D� XQD�IXQFLyQ�GH�GHQVLGDG�SDUD�OD�YDULDEOH�DOHDWRULD� ;

Page 29: Variables aleatorias reales

E� H�VHU�DVt��GHWHUPLQDU�OD�IXQFLyQ�GH�GLVWULEXFLyQ�SDUD� ;F� 'HWHUPLQDU�VX�YDORU�HVSHUDGR�\�VX�YDULDQ]DG� &DOFXODU�� �� ����3 ; �� � � ����3 ; �

�����8QD�YDULDEOH�DOHDWRULD� ; WLHQH�FRPR�IXQFLyQ�GH�GHQVLGDG��

�� �

���������HQ�RWUR�FDVR

VL D [ EI [ E D

'HWHUPLQDU�D� O�YDORU�HVSHUDGRE� D�YDULDQ]D�GH�OD�YDULDEOH�DOHDWRULD� ;

Page 30: Variables aleatorias reales

�����3DUD�HO�HVSDFLR�PXHVWUDO�FRQIRUPDGR�SRU�ORV�SRVLEOHV�UHVXOWDGRV�GH�ODQ]DU�XQD�PRQHGD�WUHV�YHFHV��VH�GHILQH�XQD�YDULDEOH�DOHDWRULD�GH�OD�VLJXLHQWH�PDQHUD�

�; �1~PHUR�GH�VHOORV�HQ�FDGD�XQR�GH�ORV�SRVLEOHV�UHVXOWDGRV��

D� 'HWHUPLQDU�D� (O�UDQJR�GH� ; �

E� � ��3 ; �

F� � ��3 ; �

G� � ��3 ; �

E� 3UHVHQWDU�OD�GLVWULEXFLyQ�GH�SUREDELOLGDG�GH� ; �F� &DOFXODU� � �( ; �� � �9DU ; �\�VX�GHVYLDFLyQ�HVWiQGDU�G� 'HWHUPLQDU�OD�IXQFLyQ�GH�GLVWULEXFLyQ� � �;) [ �

�����'DGD�OD�VLJXLHQWH�IXQFLyQ

D� $QDOL]DU�VL� � �I [ �HV�XQD�IXQFLyQ�GH�GHQVLGDG�SDUD�OD�YDULDEOH�DOHDWRULD� ; �'H�VHU�DVt��FDOFXODU� � �( ; �� � �9DU ; �\�VX�GHVYLDFLyQ�HVWiQGDU�

E� 'HWHUPLQDU�OD�IXQFLyQ�GH�GLVWULEXFLyQ� � �;) [ �

�����'DGD�OD�VLJXLHQWH�IXQFLyQ�� ���VL� �

� ��������������VL� �

[H [I [

[

D� $QDOL]DU�VL� � �I [ �HV�XQD�IXQFLyQ�GH�GHQVLGDG�SDUD�OD�YDULDEOH�DOHDWRULD� ; �'H�VHU�DVt��FDOFXODU� � �( ; �� � �9DU ; �\�VX�GHVYLDFLyQ�HVWiQGDU�

E� 'HWHUPLQDU�OD�IXQFLyQ�GH�GLVWULEXFLyQ� � �;) [ �F� &DOFXODU� � �;P W �

�����6HD� ; �XQD�YDULDEOH�DOHDWRULD�FRQ�IXQFLyQ�GH�GHQVLGDG�GDGD�SRU�������������� � �� �

� � ��� �� �� �

������������� � �

VL [

I [ F[ VL [

HQ RWUR FDVR

Page 31: Variables aleatorias reales

D� 'HWHUPLQDU�HO�YDORU�GH� FE� 2EWHQHU�OD�IXQFLyQ�GH�GLVWULEXFLyQ�GH�OD�YDULDEOH�DOHDWRULD� ;F� &DOFXODU� �� ����3 ;G� 'HWHUPLQDU� � ��� � ����3 ; ;H� &DOFXODU� � �( ; �� � �9DU ; �\�VX�GHVYLDFLyQ�HVWiQGDU

�����3DUD�XQD�YDULDEOH�DOHDWRULD� ; �FRQ�IXQFLyQ�GH�SUREDELOLGDG�GDGD�SRU�

[ � � � �

3 ; [�

D� &DOFXODU� � �( ; �� � �9DU ; �\�VX�GHVYLDFLyQ�HVWiQGDU�E� 'HWHUPLQDU�OD�IXQFLyQ�GH�GLVWULEXFLyQ� � �;) [ �\�FRQVWUXLU�VX�UHSUHVHQWDFLyQ

JUiILFD�

�����6HD� ; �XQD�YDULDEOH�DOHDWRULD�FRQ�IXQFLyQ�GH�GHQVLGDG� GDGD�SRU��

� ����� �� �� �

������������� � �

[ VL [I [

HQ RWUR FDVR

D� 2EWHQHU�OD�IXQFLyQ�GH�GLVWULEXFLyQ�GH�OD�YDULDEOH�DOHDWRULD� ;

E� &DOFXODU� � ��� ����3 ;

F� &DOFXODU� � ����3 ;

G� 'HWHUPLQDU� � ��� � ����3 ; ;

H� &DOFXODU� � �( ; �� � �9DU ; �\�VX�GHVYLDFLyQ�HVWiQGDU

I� 'HWHUPLQDU�OD�PHGLDQD

�����6HD� ; �XQD�YDULDEOH�DOHDWRULD�FRQ�IXQFLyQ�GH�GHQVLGDG�GDGD�SRU��

��������� �� �

� � � ��� �� �

���������� � �

[ VL [

I [ [ VL [

HQ RWUR FDVR

D� 2EWHQHU�OD�IXQFLyQ�GH�GLVWULEXFLyQ�GH�OD�YDULDEOH�DOHDWRULD� ;

E� &DOFXODU� � ����3 ;

F� &DOFXODU� � ����3 ;

G� 'HWHUPLQDU� � ��� � ����3 ; ;

Page 32: Variables aleatorias reales

H� &DOFXODU� � �( ; �� � �9DU ; �\�VX�GHVYLDFLyQ�HVWiQGDU

I� 'HWHUPLQDU�OD�PHGLDQD


Recommended