Page 139 - الرياضيات المتقدمة كتاب الطالب الصف 12 الفصل 2

{Oz Gb Py3s b Æ kf b .&sb b Gb ^ Æ{j b {6 1.b d?Wb ç 2;N {j b X?cb f.[ gb Ÿ zB y2b

d ±Ò –³C= ± Z ±Ò ` ³ºb ?: ±Ò –^0 : ± dg±b/4 ± R=5 : ± c9 E9 M± ` P P4 ± P b
,1Vfa yb Y U1Ne
–Îb2 ± ¤^ CÕ=4= µP E9 M± ÑQ ` R= 8 ±Ò –d4= 2 ± Z ¼b ± ϱP C C<9= : `8:
_94 ÌR5 : ± F b ±Ò –³«±R7 ± c9 ³»P7 ±Ò –ÏP ± X5hÒ –^: ± ` ¼Ò –¹b b: ± E9 Ò dg±b/4 ± R=5 : ± ] ±»C< R ± ¤Ð­ C Õ 7 C F:94 »P0: ±
´±»C ±Ò –E= ÒP ± ´±»C „± d E9 Û 0 : ± ´C »P ±Ò –d6= b ± chR ±Ò –³P P ³»C< Continuous ^0 : ± “›
– – –›œ –¢ – –›œ ´C C= ± E b: : X b ± P Ô¹C ± û0 ±
˜C R= Ò –«C Q ± random variable ¤P Ò­ –›œ ` C ±Ò –d C ± ³P b ± –R/
¤C< »P d ± d4= 2 ± Z ¼b ± VgC0 W4 ÎC: „± E C E ±¹ ˱R „±Ò E4 ±R ±

˜d C ± X b ± Îb ^ C: y Probability density d C ± X b ± ­ E b: : Ô»C=4: ±
˜C< ÒP EjR ´¹±¼ –d C ± X b ± ` _=7 ± F R ± C:9 y function CÕ R7 ’ Ô»C=4: ± ˱R „±Ò ` C ± ² ˜´C C= ± `
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˜C< ÒP EjR FÛ9 –d C ± X b ± ` _=7 ± ´P4 ± C:9 y “E b;4 ÏC »­  ²R ­ c ¯ H C; ±
d4= 2 ± c; ;: ±
zNy Fa vj&jfa q c> fa z r:Na 1yR fa ›—¢ Normal curve d C: „± Z ¼b ± ÑC ¹­ ÎÒP ± `Û= “œ _= D 8 Ô¹C ± û0 ±

Continuous random variable and the normal curve d4= 2 ± Z ¼b ± ¤½ ^06;: ± dg±b/4 ± R=5 :9 dg±b/4 ± R=5 : ± ³P b ± –R/

¹±P M± E b: c9 E:= Ô­ Q ? “½’ dg±b/4 ± R=5 :9 E;8:: ± _=7 ± F C ±º¯ Normal distribution Ÿ ž ›  œ ½ D Ò –^06;: ± E4 C ±
´„C: „±
ž
œ
Õ
continuous Õ 0 C Õ =g±b/ ±R=5 R 4 “½’ ÐA –C<; E=gS E b: Ò­ E=7=7 ± Parameters T= C7: ± ͝ Í œ › › “½’ ˜a E794 : ±
.random variable ˜½ dg±b/4 ± R=5 : ± _= D ± ­
d4= 2 ± R=5 : ±
V Îb P Э CÕ;8: T= º¯ –Îb2 ± b ^0 : ± dg±b/4 ± R=5 : ± c9 E9 M± P ­ Ô»C=4: ± ˜Í E:= P Ò­ ²
¶
˜R := ; ²R ­ c ¯ CÕ R7 a b d24 Э `8: C;;8 Ò –E= C; E P C Standard normal “Ÿ ½ ) D ±
Û
³R 6 ± `:h Z7 Îb2 ± Э d;4 ±Q< –_ ›  c24: ± Îb2 ± ÐC ±º¯ – Õ :
ÎÐÖ variable ÎÐÕ
˜_ › IŸ Îb2 ± › œIŸ E »C=4: ± ¦zNy Fa Ox2r a 40-i . fa
F C ±ºA ˜ Õ 0 CÕ=g±b/ ±ÕR=5 ^Û : ³¹„b ± G P ^62 ± E9 ÐA ^ : C Ò Standardising ˜C E R ` «S »±P7 ½C= CÕ C= ­ ]= ¯ D92 a A –E=:94 ± ÃÒR6 ± ` CÕÛ ­ F »¹ ±º¯
³R 6 ± `:h Z7 a 9 Э d;4 d< –_ Iœ ³¹„b ± G P `Û=4 ^6 E9 “¼’ “E:=7 ±’ E »P ± E4= `8 C:< Ò ˜C R= Ò­ –_ Ò­ –E C Ò­ –E9 Ò­ –` ¼ ½C= »±P7: ± Ðb8 P

˜_ IœŸ E9 8 ± I›Ÿ Z-score d;4 E90 : ± R ¹C7: ± E4= 2 –C ?2 ÀR4 E90 : ± R ¹C7: ± ½C= ÐA –E R ±
½C=7 ± ´±Ò¹­ d `:8 E P ± ÏP Ò –] º C; ÒC C:< E P C< C= c9 ³»P7 ± ÏP
R=5 : ± _= ` û9 d Ò –E P ± ` E »¹ ²R M ^0 : ± dg±b/4 ± R=5 : ± _= c24
˜E= ±P0: ± ÏP ` ±Õ»±P7 C<= ¯ û=1 G= –E R/ ± C; 4= d Ò –E P : ±
¹P4 ± c24 Э `8:= –³RgC2 ± ` c9 ` R C : ± ¹P C Q ­ ±ºA ˜^06;: ± dg±b/4 ±
˜±Q8 Ò ˜˜˜ – –œ –› –š ^ E9 C E P N R: ± ` Ò –³R= 8 ± «C2 M± ` R ­ E b ³R=50 ± «C2 M± Ðb8 P –E= C E< `
ÐA –´C C=7 ± »±R8 P; Ò ˜c ¹‡ Ò­ c9 ‡ «±b –´C C=79 R P7 «b C; P Ðb8 Э
³R 6 ± c9 E=g±b/4 ± ´±R=5 : ± ` `=Ø b; ± ` ØQ< E;8:: ± _=7 ± E= Š± ´C22 : ± ÀR4
ÀR 6 Э C; \ ±Q –R60 ± ` CÕ R ?2 ± ÎP4 Ðb8 Ò –C Õ 14 C<14 d59 «C2 M±
¤² c ¯ ­ ` “ÎC : ±’
˜«C2 M± ` CÕ R7 ÎC ´C C=7 ± ÎP4 Э
ﻞﺼﺘﻣ ﻲﺋاﻮﺸﻋ ﺮﻴﻐﺘﻣ ﻞﺼﻔﻨﻣ ﻲﺋاﻮﺸﻋ ﺮﻴﻐﺘﻣ
_ P7 E P : ± E7 R2 ±Ò –^0 : ± dg±b/4 ± R=5 : ± ³R86 E P7 ³P b ± ÑQ ÛP4Ù
أ ب أ ب ÔQ ± b Ò –^0 : ± dg±b/4 ± R=5 : ± ` P ±Ò Ãb c9 S= R ± _ = ˜a d C: „± Z ¼b ±

_=7 ± Z=: E C C;;8: –^06; dg±b/ × R=5 : d C: „± Z ¼b ± d Э CÕ7 C F »¹ ˜d4= 2 ± Z ¼b ± Z
˜E= C= ± ³P: MC X2 d Ò­ ÎÒP d Ðb8 C<; Û^8 R C;: ± ÎC: „±Ò –E;8:: ± λC d C: M± dhC R ± _ C4 ± R/ ` C ± ÐR7 ± R ±Ò­ d d4= 2 ± Z ¼b ± û/ ± P7
`8: `8 Ò –_=7 ± Z=: E C C;;8: –^0 dg±b/ R=5 : d C: „± Z ¼b ± d C ­ ´±P C/: ± d ´C C=7 ± «C2 ­ d a Î “Carl Friedrich Gauss’ ½ÒC U »P R

d R<3 d ± ´±R 6 ± ÑQ ´„C: ±Ò –´±R ^8 c9 _=7 ± ^8 ^ C8 ± ÎC : ± E C R M± d X b ± ` E R7 ± _=7 ± Э d `:8 d4= 2 ± Z ¼b ± VgC0 W4 ˜E=896 ±
.Ô»±R8 ¶»P d Ò­ –ÎÒP c; ;: ± Ð­Ò –^ ­ C< ÒP ¿R F j­ –X b ± ` _=7 ± ´P4 ± C:9 a ­Ò –C Õ = R

˜X b ± E:= Îb ^ C:

11-038 MOE book 33.indd 138 25/12/2023 4:36 PM