룅싊럖늼쯦믺쫂볾쯹캽쯦믺쫂볾쫇횸쒳튻쫂볾퓚튻듎맛달훐뿉쓜돶쿖ꎬ튲뿉쓜늻돶쿖ꎻ뿉쓜헢퇹돶쿖ꎬ튲뿉쓜쓇퇹돶쿖ꆣ죧뇈주뗄쪤뢺ꆢ돶짺펤뛹뗄탔뇰뗈ꆣ
룅싊럖늼쯦믺뇤솿벰웤럖늼튻ꆢ쯦믺뇤솿냑뿉퓚쿠춬쳵볾쿂훘뢴뷸탐ꎬ뛸퓚쫂쿈늻쓜좷뚨웤뷡맻뗄쫔퇩돆캪쯦믺쫔퇩ꆣ튪뛔쯦믺쫂볾쯹릹돉뗄쯦믺쿖쿳뷸탐짮죫퇐뺿ꎬ퓲탨뛔쯦믺쫔퇩뷡맻룸틔뚨솿쏨쫶ꆣ춳볆톧훐폃쯦믺뇤솿뛔쯦믺쫂볾뷸탐쏨쫶ꎬ늢ퟷ캪뷸튻늽췆뛏뗄믹뒡ꆣ
룅싊럖늼뛾ꆢ쯦믺뇤솿뗄쳘헷ꋅ쯦믺뇤솿뗄좡횵쫇쯦믺뗄ꎬ쫂쿈믲쫔퇩잰뚼늻횪뗀좡쓄튻룶횵ꎻꋆ쯦믺뇤솿좡쒳튻좷뚨횵뗄룅싊듳킡쫇좷뚨뗄ꆣ튻냣폃XꆢYꆢZ뇭쪾쯦믺뇤솿ꎬ폃xꆢyꆢz뇭쪾쯦믺뇤솿뗄좡횵ꆣ
룅싊럖늼죽ꆢ쯦믺쿖쿳뗄럖샠1ꆢ샫즢탍쯦믺쫔퇩쫇횸쫔퇩뗄튻쟐뿉쓜뷡맻횻폐폐쿞룶믲뿉쇐룶ꆣ죧ꎬ튻엺쇣볾훐듎욷뗄룶쫽ꆣ쯼쏇폐죧쿂탔훊ꎺꋅ믹놾쫂볾ퟩꚸ훐횻폐폐쿞룶믲뿉쇐룶믹놾쫂볾ꎬ볇캪Ω=(w1w2Lwn)
룅싊럖늼ꋆ퓚튻룶믹놾쫂볾wi훐ꎬ웤쿠펦뗄룅싊쫇룸뚨뗄ꎬ틲듋ꎬ룃쯦믺쫔퇩뗄튻쟐쯦믺쫂볾뗄룅싊뿉뇭쪾캪{p(w1)p(w2)Lp(wn)}2ꆢ솬탸탍쯦믺쫔퇩쫇횸쫔퇩뗄튻쟐뿉쓜뷡맻늻쓜폃폐쿞룶믲뿉쇐룶믹놾쫂볾뇭쪾ꆣ죧쳥훘ꆢ짭룟뗈ꆣ폫횮뛔펦뗄쫇샫즢탍쯦믺뇤솿뫍솬탸탍쯦믺뇤솿ꆣ
룅싊럖늼쯄ꆢ샫즢탍쯦믺뇤솿벰룅싊럖늼1ꆢ뚨틥퓚샫즢탍쯦믺쫔퇩훐ꎬ믹놾쫂볾ퟩꚸ훐뗄죎뫎튻룶믹놾쫂볾wi뚼캨튻뗘뛔펦떽튻룶쪵쫽횵f(w)ꎬ퓲돆쪵뇤솿f(w)ii캪샫즢탍쯦믺뇤솿ꎬ춨뎣폃XꆢYꆢZ뇭쪾ꎬ폃xꆢyꆢz뇭쪾쯦믺뇤솿뗄좡횵ꆣ죧맻튻룶쯦믺뇤솿뗄좡횵캪폐쿞룶믲뿉쇐룶쫽횵ꎬ퓲돆웤캪샫즢탍쯦믺뇤솿ꆣ
룅싊럖늼샽ꎺ뫐훐폐10룶쟲(뫬ꎭ5ꆢ싌ꎭ3ꆢ삶ꎭ2)ꆣ죎좡튻쟲ꎬw볇=뫬쟲wꎻ=싌쟲w123ꎻ=삺쟲ꎬ웤퇕즫뗄믹놾쫂볾ퟩΩ=(w1w2w3)캪ꆣ늢뿉f(틽w)죫튻룶쪵몯쫽ꎬ퓚죽훖쟩뿶쿂럖뇰좡횵+1ꆢ0ꆢ-1ꆣ볇캪+1w=w1fw()=0w=w2−1w=w3
룅싊럖늼2ꆢ샫즢탍쯦믺뇤솿뗄룅싊럖늼샫즢탍쯦믺뇤솿뗄좫늿뿉쓜좡횵벰웤쿠펦룅싊돆캪샫즢탍쯦믺뇤솿뗄룅싊럖늼ꆣ짨X 캪샫즢탍쯦믺뇤솿ꎬ좡횵캪x1ꎬx2ꎬLꎬxnꎬL쿠펦룅싊횵캪f(x1)f(x2)Lf(xn)L
룅싊럖늼웤훐ꎬf(x)=P(X=x)i=1ꎬ2ꎬ…ꎬnꎬ…iiX 뗄뿉쓜좡횵폫쿠펦뗄룅싊쇐뇭죧쿂ꎺX x 1 x 2 …xn…f(x)f(x1)f(x2)…f(xn)…돆듋뇭캪쯦믺뇤솿뗄룅싊럖늼뇭ꆣ
룅싊럖늼샫즢탍쯦믺뇤솿뗄룅싊럖늼싺ퟣ틔쿂쳵볾ꎺf(x)≥0ꋅif(xi=1ꎬ2ꎬ…ꎬnꎬ…1)+f(x2)+L+f(xn)+L=1ꋆxi벴샫즢탍쯦믺뇤솿X 좡룷룶쫽횵뗄룅싊횮뫍캪1ꆣ
룅싊럖늼샽ꎺ10 볾닺욷훐폐2 볾듎욷ꎬ죎좡2 볾ꎬ듎욷뗄룅싊캪ꎺP=P(X=x)=(Cx−ii8⋅Cii2)C210P0= P(X=0) = 28/45P1= P(X=1) = 16/45P0= P(X=2) = 1/45X=xx=0 x=1 x=2iiiiP(X=x)=P28/45 16/45 1/45iiP(Xꇜx) 28/45 44/45 1i
룅싊럖늼샽ꎺ훀솽듎펲뇒ꎬ맛달돶쿖헽쏦뗄듎쫽ꆣ믹놾쫂볾ퟩꚸ= (ꛘ1ꎬꛘ2ꎬꛘ3)웤훐ꎬꛘ1 = “늻돶쿖헽쏦”= (랴ꎬ랴)ꛘ2 = “돶쿖튻듎헽쏦”= (헽ꎬ랴)+(랴,헽)ꛘ3 = “돶X쿖2듎헽쏦”= (헽ꎬ헽)0 1 2i F(x) 1/4 1/2 1/4i
룅싊럖늼떱ꎭꇞ< x< 0쪱ꎬP(Xꇜx)=P(ꛕ)= 0떱0 ꇜx< 1쪱ꎬP(Xꇜx)=P(X=0)=1/4떱1 ꇜx< 2쪱ꎬP(Xꇜx)=P(X=0)+P(X=1)=3/4떱2 ꇜx< +ꇞ쪱ꎬP(Xꇜx)=P(X=0)+P(X=1)+P(X=2)=1
룅싊럖늼뚨틥ꎺ짨X 쫇튻룶쯦믺뇤솿ꎬ퓲돆몯쫽F(x)=P(Xꇜx) ꎭꇞ< x< +ꇞ캪쯦믺뇤솿X 뗄럖늼몯쫽ꆣ퓚짏샽훐ꎬ0−∞<x<01/40≤x<1F(x)=3/41≤x<212≤x<+∞
룅싊럖늼3ꆢ뛾뗣럖늼떱쿠뮥뛀솢뗄훘뢴쫔퇩횻폐“돉릦”뫍“쪧냜”솽훖뷡맻쪱ꎬ헢훖쫔퇩돆캪놴얬샯쫔퇩ꎬ뿉좡1돉릦X=0쪧냜
룅싊럖늼쯼뿉폃P(X=1) =p뫍P(X=0) =1-p =q뇭쪾ꎬ웤룅싊럖늼캪px=1P(X=x)=1−px=0
룅싊럖늼컥ꆢ솬탸탍쯦믺뇤솿뗄룅싊럖늼1ꆢ솬탸쯦믺뇤솿뗄럖늼쏜뛈뚨틥ꎺ솬탸탍쯦믺뇤솿X퓚(ꎭꇞꎬꎫꇞ)쓚좡횵ꎬ죧맻듦퓚뿉믽몯쫽f(x)ꎬ쪹X럾듓룅싊맦싉xP2(x1≤X≤x2)=∫f(x)dxx1,x2∈(−∞,+∞)x1퓲돆f(x) 캪X 뗄럖늼쏜뛈ꆣ
룅싊럖늼솬탸쯦믺뇤솿뗄탔훊ꎺꋙf(x)≥0ꋚx2∫f(x)dx=1x평폚쯦1믺뇤솿뗄좡횵쫇솬탸뗄ꎬ늻뿉쇐뗄ꎬ웤럖늼쏜뛈쓑폚폃뇭뇭쪾ꎬ떫뿉폃춼뇭쪾웤럖늼맦싉ꆣ
룅싊럖늼2ꆢ럖늼몯쫽쯦믺뇤솿뗄럖늼쏜뛈쫇뛔쯦믺뇤솿좡횵벰웤뇤뮯맦싉뗄ퟮ믹놾쏨쫶ꆣ튪볆쯣쯦믺뇤솿퓚쒳튻쟸볤뗄좡횵ꎬ뻍튪샻폃럖늼쏜뛈쟳뫍ꎬ튲뻍폃떽쇋쯦믺뇤솿뗄럖늼몯쫽ꆣ뚨틥ꎺ짨X캪튻쯦믺뇤솿ꎬx캪죎틢쪵쫽ꎬ퓲돆F(x)=P(X<x)(−∞<x<+∞)캪쯦믺뇤솿X 뗄럖늼몯쫽ꎬ볇캪XꆫF(x)ꆣ
룅싊럖늼평폚솬탸탍쯦믺뇤솿퓚X=x뗣좡횵캪0ꎬ벴P(X=x)=0쯹틔ꎬP(X<x)=P(X≤x)쿔좻ꎬ럖늼몯쫽F(x) 쪵훊짏뻍쫇쯦믺뇤솿X좡횵킡폚x뗄늿럖쯹햼냙럖뇈ꆣ쯼쫇춨맽뛔럖늼쏜뛈쟳뫍뗃떽뗄ꆣ
룅싊럖늼샫즢탍럖늼몯쫽뗄볆쯣ꎺk−1F(x)=P(X<x)()k=∑fxi솬탸탍럖늼몯쫽뗄볆쯣ꎺi=1xF(x)=P(X<x)=∫f(x)dx−∞
룅싊럖늼럖늼몯쫽뗄탔훊ꎺꋙ0≤F(x)≤1ꋚF(+∞)=limP(X<x)=1x→+∞F(−∞)=limP(X<x)=0x→−∞ꋛ죴좡x1=aꎬxk=bꎬ퓲b∫f(x)dx=F(b)−F(a)믲a∑f(x)=F(b)−F(a)a≤x<bi
룅싊럖늼3ꆢ헽첬럖늼ퟮ뎣볻쟒ퟮ폐폃뗄솬탸탍쯦믺뇤솿럖늼쫇헽첬럖늼ꆣ헽첬럖늼쟺쿟쫇튻쳵쿲헽ꆢ랴솽룶랽쿲컞쿞퇓짬뗄뛔돆뗄훓탎쟺쿟ꆣ쟺쿟풽살풽뷓뷼폚쮮욽훡ꎬ떫폀늻쿠붻ꆣ쟺쿟퓚x 뗄욽뻹횵뒦폐벫듳횵ꎬx 좡횵뛔돆뗘쏜벯퓚웤훜캧ꆣ
룅싊럖늼ퟮ룟뗣맕뗣ꛒ맕뗣ꛌ-ꛒꛌꛌ+ꛒx헽첬럖늼춼
룅싊럖늼헽첬럖늼뗄룅싊쏜뛈21x−m−1⋅()=2sfx⋅e2p⋅s볲볇ퟷXꆫN(mꎬs2)웤훐ꎬꛒꎭ쯦믺뇤솿X 뗄뇪ힼ닮ꎬꛒ>0ꎻꛌꎭ쯦믺뇤솿X 뗄뻹횵(쫽톧웚췻)ꆣ
룅싊럖늼ꛌ뫍s2쫇헽첬럖늼뗄솽룶쳘헷닎쫽ꎬꛌ뫍s2뗄늻춬ퟩ뫏퓲뻶뚨쇋쟺쿟뗄탎ힴꆢ캻훃닮틬ꆣ떱ꛌ=0 쟒ꛒ=1쪱ꎬ쿠펦뗄럖늼N(0ꎬ1) 돆캪뇪ힼ헽첬럖늼ꆣm1=m2s1≠s2m1≠m2s1≠s2m1≠m2s1=s2ꛌ1 ꛌ2ꛌ1,ꛌ2ꛌ1 ꛌ2
룅싊럖늼헽첬럖늼쫇ퟮ뎣볻뗄럖늼ꎬ듳솿뿍맛쿖쿳럾듓믲뷼쯆럾듓헽첬럖늼ꆣ죧죋뗄쳥룟ꆢ쳥훘ꆢ닺욷탔쓜뗈ꆢ뗧ퟓ맜뗄죈퓫짹ꆢ쿋캬잿뛈ꆢ솸쪳닺솿ꆢ닺욷쫙쏼뗈ꆣ헽첬럖늼뻟폐뫜뫃뗄쫽톧탔훊ꆣꋙ탭뛠럇헽첬럖늼틔헽첬럖늼캪붥뷼럖늼믲벫쿞럖늼ꎻꋚ죎틢퇹놾뻹횵뗄럖늼뚼틔헽첬럖늼캪벫쿞럖늼ꎬ헢뻍쫇룅싊싛뗄훐탄벫쿞뚨샭쯹횤쏷뗄ꆣ
룅싊럖늼뻟쳥뗘쮵ꎬ떱ퟜ쳥쫇헽첬럖늼쪱ꎬ늻싛퇹놾죝솿듳킡ꎬ퇹놾뻹횵뚼럾듓폚헽첬럖늼ꎻ떱ퟜ쳥늻럾듓폚헽첬럖늼쪱ꎬ횻튪퇹놾죝솿ퟣ릻듳ꎬ퇹놾뻹횵뗄럖늼튲쟷뷼폚헽첬럖늼ꆣ
룅싊럖늼헽첬럖늼뗄쳘헷ꎺꋅ헽첬럖늼쏜뛈틔x=ꛌ캪뛔돆훡ꎬ늢퓚x=ꛌퟮ듳횵ꎻꋆ떱xꇺꇀꇞ쪱ꎬf (x)ꇺ0ꎬ맊f (x) 틔x 훡캪붥뷼쿟ꎻꋇ쟳떼횪ꎬx=ꛌꇀꛒ캪뗄솽룶맕뗣ꎬꛒ캪맕뗣떽뛔돆훡ꛌ뗄뻠샫ꆣ죧맻ꛒ뷏듳ꎬ퓲쟺쿟뷏욽뮺ꎬ랴횮ꎬ쟺쿟뷏뚸쟍ꆣ
룅싊럖늼헽첬럖늼x=ꛌ뛔돆훡ퟳ폒쟺쿟쿂뗄쏦믽룷캪1/2ꆣP(Xꇜꛌ) = (Xꇝꛌ) = (ꛌ-ꛒꇜXꇜꛌ+ꛒ) = (ꛌ-2ꛒꇜXꇜꛌ+2ꛒ) = (ꛌ-3ꛒꇜXꇜꛌ+3ꛒ) =
룅싊럖늼퓚헽첬럖늼훐ꎬ뾿뷼뻹횵훜캧돶쿖뗄뇤솿욵쫽뷏룟ꎬ웤뇤솿횵실퓚ꛌꇀ3ꛒ췢뗄룅싊짙폚1% (‰)ꎬ헢쫇튻룶뫜킡뗄룅싊ꎬ뻍튻듎쯦믺쫔퇩믲맛달살붲ꎬ쯼랢짺뗄룅싊쪵퓚첫킡쇋ꎬ폐샭평죏캪룃뇤솿횵늻쫇살ퟔ헢튻헽첬럖늼ꎬ뛸쫇평웤쯻쾵춳탔풭틲쯹훂ꆣ헢뻍쫇킡룅싊풭샭믲3ꛒ풭샭ꎬ놻맣랺펦폃폚닺욷훊솿뿘훆ꆢ닺욷훊솿볬퇩뫍죕뎣짺믮훐ꆣ
룅싊럖늼4ꆢ뇪ힼ헽첬럖늼돆XꆫN(0ꎬ1) 캪뇪ힼ헽첬럖늼ꆣ뛔폚럇뇪ힼ헽첬럖늼Xꎬퟷ죧쿂뇤뮻2X−m1z−Z=f(z)=2⋅es2p퓲뿉붫웤뇪ힼ뮯캪ZꆫN(0ꎬ1)
룅싊럖늼뇪ힼ헽첬뇤솿뗄럖늼몯쫽F(x) 캪21xxx−F(x)=()2∫fx⋅dx=⋅e⋅dx−∞∫−∞2p폐맘헽첬럖늼뗄쪵볊컊쳢튻냣뚼뮻캪뇪ힼ헽첬뇤솿뗄럖늼컊쳢뫳뷸탐볆쯣ꆢ럖컶ꆣ틲듋ꎬ뇪ힼ헽첬럖늼퓚쪵볊펦폃훐뻟폐훘튪뗘캻ꆣ
룅싊럖늼헽첬뇤솿X 퓚x1뫍x2횮볤좡횵뗄룅싊뗈폚뷸탐뇪ힼ뇤뮻뫳뇪ힼ헽첬뇤솿Z 퓚쟸볤((x1−m)sꎬ(x2−m)s)짏좡횵뗄룅싊ꆣ벴P(x1< X< x2) = P(z1< Z< z2)웤훐ꎬx1−mx2−mz1=z2=ss
룅싊럖늼샽ꎺ죴XꆫN(5ꎬ22) ꎬ볆쯣P (2 < X < 3)ꆣ쿔좻ꎬx1=2ꎬx2=3ꎬꛌ=5ꎬꛒ=2ꆣ쇮x1−m2−5z1===−−m3−5z2===−1s2틲듋ꎬP (2 < X < 3) = P ( < Z < -1)
룅싊럖늼뛸P ( < Z < -1) = P (1 < Z < )= = %쿔좻ꎬP (2 < X < 3) = %
룅싊럖늼5ꆢ헽첬럖늼쏦믽뇭뗄펦폃샽ꎺ볆쯣P ( < Z < 0)ꆣ뷢ꎺ평뛔돆탔횪ꎬP ( < Z < 0) = P (0 < Z < )뛸P ( Z ꇜ0) = ( Z < ) = (0 < Z < ) = P ( Z < ) -P ( Z ꇜ0) = 벴P ( < Z < 0) =
룅싊럖늼샽ꎺ볆쯣P (Z > )ꆣ뷢ꎺP (Z > ) = 1-P (Z ꇜ)= = 샽ꎺ볆쯣P (Z < )ꆣ뷢ꎺ평헽첬쟺쿟뗄뛔돆탔횪ꎺP (Z < ) = P (Z > )-z00z0= 1-P (Z ꇜ)= =
룅싊럖늼샽ꎺ볆쯣P (< Z < )ꆣ뷢ꎺP (< Z < )= P (Z < ) -P (Z ꇜ)0z0z1= = 샽ꎺ볆쯣P ( < Z < )ꆣ뷢ꎺP ( < Z < ) = P ( < Z < )= P (Z < ) -P (Z ꇜ) = –= =-z1-z000z0z1
룅싊럖늼샽ꎺ볆쯣P (< Z < )ꆣ뷢ꎺP (< Z < )= P (Z < ) -P ( Zꇜ)= P (Z < ) –(1-P ( Z < ))=–() =
룅싊럖늼6ꆢ뛔쯦믺뇤솿X 뗄뇪ힼ뮯뇤뮻뛔폚튻냣뗄헽첬럖늼컊쳢ꎬ탨튪쿈뮻캪뇪ힼ헽첬럖늼컊쳢ꎬ좻뫳퓙뷸탐볆쯣ꆣ샽ꎺ쒳웳튵맘볼릤탲릤죋뗄ퟷ튵쪱볤럾듓헽첬럖늼ꎬ뻹횵캪20 럖훓ꎬ뇪ힼ닮캪3 럖훓ꆣ볆쯣ꎺꋅ죎톡튻죋ꎬퟷ튵쪱볤짙폚17럖훓뗄룅싊ꎻꋆ쓜럱틔95 %뗄룅싊놣횤룃릤탲짺닺쪱볤늻뎬맽25럖훓ꎿꋇ죴내10 %뗄뇈샽붱샸킧싊ퟮ룟뗄릤죋ꎬ펦죧뫎훆뚨붱샸ퟷ튵뇪ힼꎿ
룅싊럖늼뷢ꎺ틑횪XꆫN(20ꎬ32)ꎬꛌ=20ꎬꛒ=3쇮X−mX−20Z==s3ꋅPX17−20(<17)=PZ<3=P(Z<−1)=F(−1)=1−F(1)=죎톡튻죋ꎬퟷ튵쪱볤짙폚17럖훓뗄룅싊캪%
룅싊럖늼ꋆ25−20PX(<25)=PZ<3=P(Z<)=F()=쿔좻ꎬ튪쟳ퟷ튵쪱볤틔95%뗄룅싊늻뎬맽25럖훓뗄튪쟳뿉틔싺ퟣꆣꋇx−20PX(<x)=PZ<=10%3닩뇭뗃z = ꎬ뿉횪x = 펦붱샸ퟷ튵쪱볤늻뎬맽16럖9쏫뗄릤죋ꆣ
룅싊럖늼샽ꎺ틑횪XꆫN(7ꎬ52)ꎬ쟳P (1< X < 14)뷢ꎺ틑횪ꛌ=7ꎬꛒ=5ꎬx1=1ꎬx2=14쇮X−mX−7Z==s5뛔X 뇪ힼ뮯뗃x1−m1−7z1===−−m14−7z2===
룅싊럖늼P (1< X < 14)= P (< Z < )= P (Z < ) -P ( Z < )= P (Z < ) –(1-P ( Z < ))= –()= –=
룅싊럖늼5ꆢ뻹퓈럖늼쯦믺뇤솿X 퓚폐쿞쟸볤(aꎬb)쓚좡횵ꎬ쟒웤룅싊럖늼캪1(b−a)a<x<bf(t)=0웤쯻퓲돆X 퓚쟸볤(aꎬb)짏럾듓뻹퓈럖늼ꆣ
