ESTAD©iSTIcA B ©ASIcA Variable aleatoria: es una variable x que

!"#$&%' (*),+.-/"0213054768059:;)<!=#-)*>-),?)A@CBEDF034HG,-"05:I=J-6;)K4L1305:;B54M),/N-/N6I/#@C)K4PO
1305:;BRQTS"UWVYX[Z
]\_^`#a
Jbc$%d(eGKB5/+C6I+7@Y)f),/-/0HDH)K?6I?"0g?)f-/"0h13054C6;059:I)i05:;)K03@YB54C680jZ
kmln$popqr#'% btsdu^pv%wxmsd"szy'{e| <c}~(+7)?)n€"/)?)DM05/),4Y0L=#-):;0H>4CBE9"0396;:I6;?"05?F?)=#-)3y
05:c"05GK),4‚-/k)nƒj>')K4C6IDM),/r@YBh+CB594C)„<dy/BE+‚+Y05:I…r0-/1305:IBE4†),/R),:6;/r@C)K471305:IBQ <dUC<‡‰ˆE<"X[y),+Š{e| <c}‹ˆE<dZ
Œ -R6;/r@Y),…E4Y03:)K/R)K:_6I/#@C)K4‹1503:;Bh?)f?)n€"/6;G,6EB /k?)i<A"0h?)+7)K4Ž:80g-/6;?"03?y
H ˆE<c{e|’<c}`“•”
‘
– 0H=#-):;0L>4CBE9"0396;:I6;?"05??)=#-)<u+Y05:I…r0h)K/*),+C)6;/r@Y),471305:IBH),+]-/BZ Œ )?6;G,)f)K/r@YBE/GK)K+]=J-)h{e|’<c}
)K+‹@w.0 —™˜š~›Hœ5Ÿž¡ œr¢rœ5Ze£d0g>47BE9"0596;:;6I?"05?H?)i=#-)]<+Y05:I…r0),/k)K:™6I/r@Y)K4‹1305:;B.Q¤S"UC<"XmyGKB5/kSF¥¦<A¥§Vy)K+,(
¨F|’<c}©“ Lª ˆE<"«¬{e|’<"«¬}
‘
Œ 6x{e|’<c}†/BH)K+7@"H0 /BE4CDM05:;6I­&05?0yj),/r@YBE/G,)K+
 ª ˆE< « {e|’< « }
¨F|’<c}©“ H‘ 
ˆE< « {e|’< « }
‘
® 9r1j6;05DH)K/r@Y)¨M|mSJ}`“°¯ – ¨M|mVn}`“•”EZ
±²%'a
Jbc$%'lLsxu ³spl,$ oxqr#c% bz(+7)?),€"/),/AG,BEDMB
<´¶µ·d¸º¹’<"µ#»7¼]“ H ˆE<"<"µE{e| <c}~U ½“¾”EU~¿jUWÀjU&ÁIÁ¡Á
‘
Œ 6x{e|’<c}†/BH)K+7@"H0 /BE4CDM05:;6I­&05?0yj),/r@YBE/G,)K+
H ˆE<"< µ {| <c}
<´Âµ&·d“ ‘ H
U ½“Ô5U~¿jUWÀUKÁIÁ¡Á
ˆE<c{e| <c}
‘
Ä`:`GKBE/BjG,6;DH6;)K/r@CB?)H@CBj?BE+i:;BE+iDHBEDH)K/r@YB5+?)H-/"0R?6;+7@C4C6I9-GK6&EB /u)K+)K=#-6¡1305:;),/#@C)H0kG,BE/BJGK),4:;0
?6I+7@Y476;9-GK65B /Z`Ä`:_>4C6IDM),4DHBEDM),/r@YBL)K+Ž:80gDH)K?680(
< ´¡Åm· “•¹<c» ¼
Ä`:_+C),…E-/?Byw< ´ÂÆP· y"+C)f>-),?)47)K:;05GK6IBE/"054G,BE/R:8013054C6;05/­K0LÇ ª (
Ç ª Æ “•È#| <.ɧ¹<c»Y} Æ&Ê ¼ “•È&< Æ Ê ¼ É˹’<c» Ƽ “•È&< Æ Ê É!¹<c» Æ
g" #$%wFspl&qEvr$(d)K/h),+7@Y)‚GK05+CBi:;0Š13054768059:I)†03:;)&0@YBE47680‚<M>-)K?)@CBEDM054pG,-"05:I=J-6;)K4C0
?)f-/R/„- DH)K4CBh?6;+CG,4C)n@YBH?)i1503:;BE47)K+Ky
<“ÃÌ< Å UC< Æ U&Á¡Á¡ÁIUC<'͂Î
”
£d0*Ï -/GK6&EB /Ð?)k>47BE9"0596I:;6I?"05?
),+H05BE4C0Ñ{vÒÓy©=#-)k+C)?)n€"/)kGKB5DMBÑ:80>4CBE90596;:I6;?"03?‰?)=#-)k-/
)nƒj>')K476;DH)K/r@YBL+CB594C)f<*47)K+C-:I@Y)f),/-/k1305:;B54†<"ÒÓZ©£d0h/BE4CDM05:I6;­&03GK65B /F)K+,(
Í
Ô ¨dÒc“•”
Ò¬Õ Å
– :IBE+‚DMB5DM),/#@CBE+‚+C)f?)n€"/),/ADH)K?6;05/r@Y)5(
Í
Ô
<"µÒ ¨xÒ
Í
¬
Ò
Õ
<´¶µ·d“ Ô <"µÒ ¨xÒ[U <´¶µ·“ Í Å
Ò¬Õ Å
Ô ¨dÒ
Ò¬Õ Å
£d0h+C)K…5-/?"0H)nƒj>4C),+C6&EB /+7)05>:;6IG&0g+76_:80h?6;+‹@Y476;9-G,6EB //BH),+7@"H0 /B54CDM05:;6I­&05?"0jZ
kmln$popqr %'bxJlga
^`%w$"bc$&jl
Ö"× Rl,$ oxqr#c% bØpmbd%'a‰zÙtqE"a‰bx%³" #$%w%
Œ )K0M)K:d+76;…E-6I)K/r@Y)>47BE9:;),DF0j(©-/"0M>"0347@&Ú  GK-:;0H+7)>-)K?)?)K+7>:803­&054Û0M:IBH:;054C…5BM?)-/"0H?6;DH)K/+C65B /
?"05/?Bf>"05+CB5+©?)‚:;BE/…56I@Y-?hÜÝZ Œ )&0ÞH:80„>4CBE9"0396;:I6;?"05??)‚=#-)Ž?†)Ž -/L>"05+CB0i:80Û?)K47)KGW"0F|ßf“•”_ÉÞ
+C),4"L0 :80g>47BE9"0596;:;6I?"05?.?)=#-)f?)f -/R>"05+7BF0h:;0g6;­K=#-6I)K47?"0r}nZzà„),+C>-)K +Š?)áâ>05+CBE+,y")K:_?),+C>:;05­&0vO
DH6;)K/r@CB]@YB5@Y05:J+C)K4„0 6;…E-05:0ÛãkÜPyr?BE/?)„ÉÛáâä§ãåä!á2Z Œ )&0„½ Å )K:/]- DM),4CB„?)†>"05+CB5+`0„:80Š?)K47)KGW"0
@Y4C05+Ž)K+7BE+Šáâ>05+CBE+,y – ½ Æ )K:_/]- DH)K47Bh?)f>"05+7BE+Š0H:;0g6;­K=#-6I)K47?"0ZzÄ`/r@YBE/G,)K+Š½ Å ‡æ½ Æ “°á³Z
ç`-)K+‹@YBR=#-)LG&05?"0k>"05+7Bk)K+f6;/?)K>')K/?6I)K/r@Y)L?),:z05/r@Y),4C6;B54Ky_:80.>4CBE9"0396;:I6;?"05??)L=#-)L:80.>"054‹@Ú  GK-:;0
"05…E0g-/"0h+C),GK-),/GK6;0HG,-"05:;=#-6I)K4C0g?"05?"0h?)i½ Å >"03+CBE+Ž0h:80g?)K4C),GW"0 – ½ Æ >"05+7BE+Ž0h:806I­K=#-6I)K4C?0L),+
ÞµèPßµ&é
çp),4CBH"0 –
á2ê
½ Šꤽ Æ ê
DM05/)K4C05+†?6;+7@C6;/r@W03+†?)„47)&05:I6;­K054½ Å >"05+7BE+Ž0:;0?),4C)KGW0 – ½ Æ >05+CBE+0:806;­K=#-6I)K47?"0Zz£d0>47BE9"0596I:¡O
6;?05?u?)h=#-)h:80M>"0547@&Ú  G,-:80F?)h ½ Å >"05+7BE+f0.:80.?)K4C),GW"0 – ½ Æ >05+CBE+0.:;0F6;­,=J-6;)K47?"0y),/ÑGK-"05:I=#-6;),4
+C),GK-),/GK6;0yw)K+
¨M|½ Å }©“ ½ á³ê¤½ ê ê Þ µ3è ß µ&é
Å Æ
:80.:I:805DM05?"0*¢5ž;ënìmš~žíKî"ïnž"𘠗ÐíKž’—™˜›hžœ5ŸZh£d04Y05­EB /Ñ?)L)K+‹@W0k?)K/B5DM6I/"05GK6&EB /u)K+f=#-)L)K+f6I…E-"05:e=#-)L),:
GKBJ),€GK6;),/r@Y)f?)K:d?),+Y05474CBE:I:;Bh?),:_96;/BEDM6IBg?)ñÛ)nò‚@YBE/™(
Í
|ŸÞ‡æß5} Í “ Ô ½ê;|máôá³ÉÐê ½_}~ê Þµ3ß Ízõ µ
µ Õó
Œ 6d+Y039c),DMBE+Š½ yc>'Bj?)KDHBE+ŽGKBE/BJGK),4Û㠖 0M=#-)ãö“÷½ Éæ½ “¾¿3½ Éæá2Z‚£d0H>4CBE90596;:I6;?"03??)
=#-):80g>"054‹@Ú  GK-Å :;0g+C)f?)K+7>:803GK)f?)K+7?)),:_BE4C6I…E)K/05+7@Y0HãkÅ Ü),+ŠÆ ¨F| ½ Å }zÅ “ø¨M|C| ½ Å ‡¦áA}CùE¿E}~Z
Ä`:]>47BE9:I)KDM02?),:]G&03DM6I/BÐ05:I)&03@CBE4C6IB2/B
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¿
ú -"05:I=#-6;),4]>47BE9:;),DF0H)K/u),:=#-)h"0 – 0.-/u)nƒj>c),4C6IDM),/#@CB.GKB5/u-/u/„- DH)K4CBM?"05?B?)g>cB5+C6;9:;)K+Û4C)~O
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4C),+C-:¡@W05?B2@Y),/…r0³>4CBE90596;:I6;?"03?ÑÞy – @W05:Š=#-)A),:
4C),+C-:¡@W05?B
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)K+‹@W05?Ú  +7@C6;GK05DM),/r@Y)g6I/?)K>')K/?6;),/#@C)K+Y}ny_+C6I…E-)g-/"0.?6;+‹@Y476;9-G,6EB /A96;/B5DM6;05:[ZÛÄ`+i)K:eGK05+CB?),:e:803/­&0O
DH6;)K/r@CBR?)H-/"0RDHBE/),?"0yd?)H+Y05GK054f9cB5:805+i?)HGKB5:;BE47)K+i?)H-/³+Y05G,BG,BE/
47)K)KDH>:;05­KBy_?)MG,BE:;BJG&054
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ü©05DHBE+Ž0hG&05:IGK-:;054:IBE+1305:;BE47)K+‚DH)K?6IBE+KZz£d0hDH)K?6;0g)K+K(
Í
Í
á³ê
Ô
Ô
½
ý “ Õó ½™¨F| ½_}z“ Õó ½pê8|áôɉ½_}~ê ½JÞ µ ß Ízõ µ
µ
µ
þ]0 – =#-)"05G,)K4Ž/B5@Y054‚=#-)½jÞ µ “ØÞx|mÿ'ù5ÿEÞ'},|¬Þ µ }ny"GKB5/:IBhGK-"05:
Í
Í
³
á
ê
ÿ
ÿ
á³ê
z
Í
õ
Ô
Ô
½
ý “ Õó ½pê8|áôÉн_}nê Þ ÿEÞ Þµß µ“ØÞ ÿEÞ Õó ½ê;|máôÉн_}~ê Þµ3ß Ízõ µ“
µ
µ
“æ
Þ ÿEÿ Þ |ŸÞ‡æß5} Í “æÞwá
|¬Þ‡æß5} Í`õ Åz“°áHÞ
üz)&03DMBE+Ž03BE4Y0h),:_DMBEDH)K/r@CBH?)B54C?),/*¿j(
Í
Í
Í
Æ
Ƚ Æ Ê “ Ô ½ Æ ¨M|½}`“ Ô ½pê8|á á2ÉÐê ½}nê ½ Æ Þ µ ß Ízõ µ “ Ô ½ê;|máôá³ÉÐê ½_}~ê Þ ÿEÿ Þ Þ µ ß `Í õ µ
µ Õó
µ Õó
µ Õó
Æ ÔÍ
á2ê Þ µ ß Í`õ µ “ Þ ÿ Æ |¬Þ‡ØßE} Í
ÿ
“ Þ ÿEÞ ÿEÞ Õó ½pê8|á É
½_}nê
µ
“æÞxQTá‰|ŸÞg‡ØßE} Ízõ Å_‡ÑÞwá
|máôÉN”v},|¬Þ‡æß5} Í`õ ÆCXc“øáLÞg‡æáHÞ"Æ|áôɧ”v}
£d0g15034C6803/­&0L05:G,-"05?4C05?BH)K+,(
Ç ÍÆ “ËáHÞg‡¦áLÞ Æ |má ɧ”v}ɧ|máHÞw} Æ “ËáHÞMÉØáHÞ Æ “øáLÞx|‹”ŽÉ*Þ'}`“øáLÞ"ß
ÇwÍN?"0L6;?)K0h?)f:80h05/GW#-4C0M?)f:;0g?6;+‹@Y4C6I9-G,6EB /Z
× Rl,$ oxqr#c% b
w"oxl&l& "bx
ç034Y0Ðá …E4C05/?)A:802?6;+7@C4C6I9-GK6&EB /t96;/BEDH6803:Š+7)*>-),?)u)K+7GK476;96I4F?)*-/"0‰DF03/)K4C0‰05>47Bƒj6IDF05?0
>')K4CBFDN50 +ŠGKB5/#15)K/6I)K/r@Y)3ZfÄ`/*)K+‹@Y)hG&03+CB.:;0F05/GW#-4C0F?)g:;0M?6I+7@C4C6;9-GK6&EB /),+„Dg- – >')K=#-)/ 0y – :;B5+
- /6IGKBE+Ž1305:;BE47)K+Û+C6;…5/6I€"GK03@Y6¡1EBE+Š?)g½
+CBE/u03=J-)K:;:IBE+]=#-))K+7@"50 /uGK),4CGK0.?)K:pDN0 ƒj6IDMBZ Œ )K0¦)K +‹@Y)½ –
?),+Y05474CBE:I:;)KDHBE+:IBE…©¨F| ½_}05:I4C),?)K?B54‚?)f)K+7@C)Dt0 ƒj6;DHB(
:;BE…©¨M|½_}`“Ë:;BE…z¨M|½ }_‡ ˆ:IBE…zˆr½¨M|½_} |½É½_ }_‡ ¿” ˆ Æ :IBEˆ#…z½ ¨M|½_} |½É½_ } Æ ‡!ÁIÁ¡Á
Æ µ
µ
ú BEDHB:;0R?)K4C6¡1305?"0k>4C6IDM),4Y0k)K/
),:Dt0 ƒj6;DHBk )K+/#-:80y – :80k?)K476I1305?"0k+C),…E -/?0A)K+/)K…r0@Y6I1302|:;0
"05G,)KDHBE+Ž6;…E-05:0
É }ny@C)K/),DMBE+,(
:;BE…z¨M|½}ø:;BE…©¨M|½_ }É ¿” ¬|½Fɽ_ }‹Æ ¨F| ½_}`“°¨M|½ } õ é!è "# ´¶µ õ µ&· é
À
`Ä +7@Y0.)K+„:;0F:;:;05DF03?"0R¢5ž;ënìmš~žíKî"ïnž"𘠗$rœ5îrëWënžœ5—'œ5Z ú 05:IGK-:I)KDHBE+]03BE4Y0M)K:Dt0 ƒj6;DHB%½ – :;0M?),4C6¡1503?"0
+C),…E-/?0&©Z('x)K/)KDHBE+K(
:;BE…©¨M|½_}`“!:;BE…©á³êrɉ:;BE…z½pêrɉ:;BE…Û|á Éн_}nê‡æ½i:;B5…dއ°|máôɉ½_}:IBE…zß
“øá÷:IBE…záôÉØá Éнi:;BE…`½M‡Ø½ÉË|má Éн_}:IBE…Š|máôÉн}_‡¦áôɉ½M‡æ½i:IBE…Þ‡ø|máôɉ½}:;BE…zß
à„)K476I1305/?B(
ˆ:IBE…`¨M|½_} “ÃÉ*:;BE…)½H ‡Ø:;BE…Š|á É*½ }_‡æ:IBE…_ÞMÉ
:;BE…zßf“°:IBE… |má É ½ } Þ “°¯
ˆr½
½™ ß
µ
?)f?BE/?)
|á *
É ½d } Þ “•” ½
“ áHÞ “øáLÞ
½c ß
އæß
® +7)&0yw),:_DN0 ƒj6IDMBgG,BE6;/GK6;?)iGKBE/R:80gDH)K?6;0Z +ہBE4Y0j(
” “¾É ” É
”
” ” ‡ ” “¾É ” “ÃÉ ”
ѓ ˆ Æ :;BE…z¨M|½} “•É ” É
•
“
É
ˆr½ Æ µ
½ á É ½
áLÞ á ÉØáHÞ
á Þ ,ß áLÞ"ß
Ç ÍÆ
® +C)K0-y ¦),+„),Ï ),G,@C 6I 1305DH)K/r@Y)g/)K…r03@C6I15B – ),+„6;…5-"05:x05:6I/r1E)K47+CB?):80H1305476805/­&0.05:xGK-05?4Y03?BZiçxBE4
@W03/#@CB(
É ½_ }PÆ
õ |½
¨M|½_}`“°¨F.| ½ /} ¿ÇÍÆ
Ä`:_>4C)nÏ05G,@CBE4Š¨F.| ½ }+7)>-)K?)G&03:;GK-:8054)~ƒj6;…E6I)K/?BL=#-)f:80h+7-DF1
0 0 µ ¨M|½_}`“ÔE(
É ½_ } Æ
õ |½
¿ÇÍÆ
¨F| ½_}`“ |½É½ } Æ
õ
Ô ¿3Ç ÍÆ
µ
Ä`:?),+Y05474CBE:I:;Bh=#-)fGKBE/?-GK)0h:80?6;+‹@Y4C6I9-G,6EB /96I/BEDH6805: – :I-)K…5By>"034Y0g…E4C05/?),+Š/]- DM),4CBE+‚á³yw0
:80H?6;+‹@Y476;9-G,6EB /…r05-+7+C6;05/"0yc)K+iDg- – …E),/)K4C05:[Z„ñÛB)K+]+CBE47>4C),/?)K/r@C)5yd>-),+Ky=#-)h:;0M?6I+7@Y476;9-GK65B /
…r05-+C+C6;05/"0h+C)f>47)K+7)K/r@Y))K/RDg-:I@C6I@C-??)f>4CB59:;),DF05+‚),/),+7@Y05?_Ú  +‹@Y6IG&0 – )K/kÏ‹Ú  +76;GK0g)K+7@Y05?_Ú  +7@Y6IG&0Z
2