已知随机变量X~N(2,4)，且P(X<a）=0.2 ，则P（X<4-a）= ？
\u5df2\u77e5\u968f\u673a\u53d8\u91cfx\uff5eN\uff082\uff0c\u03c32\uff09\uff0c\u82e5P\uff08x\uff1ca\uff09=0.32\uff0c\u5219p\uff08a\u2264x\uff1c4-a\uff09\u7684\u503c\u4e3a______\uff0e\u2235\u968f\u673a\u53d8\u91cfX\u670d\u4ece\u6b63\u6001\u5206\u5e03N\uff082\uff0c\u03c32\uff09\uff0c\u03bc=2\uff0c\u2234p\uff08x\uff1ca\uff09=p\uff08x\uff1e4-a\uff09=0.32\uff0e\u2234p\uff08a\u2264x\uff1c4-a\uff09=1-p\uff08x\uff1ca\uff09-p\uff08x\uff1e4-a\uff09=0.36\u6545\u7b54\u6848\u4e3a\uff1a0.36
\u2235\u968f\u673a\u53d8\u91cfX\u670d\u4ece\u6b63\u6001\u5206\u5e03N\uff082\uff0c\u03c32\uff09\uff0c\u2234\u03bc=2\uff0c\u6b63\u6001\u66f2\u7ebf\u7684\u5bf9\u79f0\u8f74x=2\uff0c\u6839\u636e\u6b63\u6001\u66f2\u7ebf\u7684\u7279\u70b9\uff0c\u2234p\uff08X\uff1ca\uff09=p\uff08X\uff1e4-a\uff09\uff0c\u4e14P\uff08a\u2264X\uff1c4-a\uff09=1-2p\uff08X\uff1ca\uff09\uff0c\u2234P\uff08a\u2264X\uff1c4-a\uff09=1-2\u00d70.26=0.48\uff0e\u6545\u7b54\u6848\u4e3a\uff1a0.48\uff0e
因为X~N(2,4)，且P(X<a）=0.2 ，所以P(X<a）=Φ（a/2-1）=0.2；所以，P（X<4-a）= Φ（（4-a-2）/2）= Φ（1-a/2）=1-Φ（a/2-1）=1-0.2=0.8
绛旓細N(2,4)涓烘鎬佸垎甯冿紝鍏朵腑X鍧囧间负X=2,鐢卞叾鍥惧儚瀵圭О鎬 P(x<1)=P(x>3)=0.3 P锛坸<3锛=1-P(x>3)=1-0.3=0.7
绛旓細锛1锛夋牴鎹鎬佸垎甯冪殑姒傜巼瀵嗗害f锛坸锛=1/[鈭氾紙2蟺锛壪僝*exp{-锛坸-渭锛²/2蟽²} 鎵浠X~N锛2锛2²锛夌殑姒傜巼瀵嗗害鎶娢=2锛屜=2浠ｅ叆寰楋紝f锛坸锛=1/[2鈭氾紙2蟺锛塢*exp{-锛坸-2锛²/8} 锛2锛夋眰P{X鈮2}锛屽厛鏍囧噯鍖栵紝P{X鈮2}=P{锛圶-2锛/2鈮わ紙2-2锛/2}=...
绛旓細灏嗘鎬鍙橀噺锛X~N(2,4) 鍙樻崲鎴愭爣鍑嗘鎬佸彉閲忥細t = (x-2)/2 浜庢槸锛2<x<6 鍙樻垚锛0 <t<2 P(2<x<6) = P(0<t<2) = 桅(2) - 桅(0)= 0.9772 - 0.5 = 0.4772
绛旓細X~N(2,4)鏄鎬佸垎甯 E(X)=2锛孌(X)=4 鏁匛(3X-5)=3E(X)-5=3脳2-5=1 D(3X-5)=9D(X)=9脳4=36 鍚岀悊锛孍(Y)=0锛孌(Y)=1 鍥犱负aX-bY鏄嚎鎬х粍鍚堬紝鏁卆X-bY涔熸湇浠庢鎬佸垎甯冦侲(aX-bY)=aE(X)-bE(Y)=2a D(aX-bY)=a²D(X)+b²D(Y)=4a²+b²...
绛旓細鍥炵瓟锛X N(2,4) 鍒: EY = aEX + b =2a + b = 0 DY= a²DX = 4a² =1 a = 1/2 b = -1
绛旓細a=2 x灞炰簬姝ｆ佸垎甯冿紝姝ｆ佸垎甯冪殑姒傜巼瀵嗗害鏄互鐩寸嚎x=渭瀵圭О鐨勶紝杩欓噷渭=2 P锛圶>a)=0.5锛屾墍浠鍒氬ソ鏄湪渭杩欎釜鐐癸紝鎵浠=2
绛旓細鐢变簬X鏈嶄粠姝ｆ佸垎甯冿紝鑰孻涓嶺鎴愮嚎鎬у叧绯伙紝鎵浠蹇呭畾涔熸垚姝ｆ佸垎甯冿紙鏈夌偣鑰楁椂闂达紝鎵浠ヤ笉璺熶綘璇佹槑銆傦級锛岃屾鎬佸垎甯冪殑鏈熸湜绛変簬渭锛屾柟宸瓑浜幭冣埀2锛鎵浠(Y)=2E(X)+3=5锛岃屾柟宸甐ar(Y)=4D(X)=16
绛旓細X锝N(渭锛屜僞2)浠ｈ〃姝ｅお鍒嗗竷锛屽叾涓煎嵆姝ｅお鍒嗗竷鍧囧硷紝蟽^2浠ｈ〃鏂瑰樊锛屾墍浠ユ棰樼瓟妗堝氨鏄4
绛旓細鏍规嵁鍧囧煎拰鏂瑰樊鍏紡,鍙緱 2a+b=0 4a^2=1 姹傚嚭a=0.5, b=-1
绛旓細P锛-3.36<X<7.36)=P((-3.36-2)/4<锛圶-2锛/4<(7.36-2)/4)=P(-1.34<锛圶-2锛/4<1.34)=蠁锛1.34锛-蠁锛-1.34锛=蠁锛1.34锛-锛1-蠁锛1.34锛夛級=2蠁锛1.34锛-1=0.82 灏辨槸涓涓爣鍑嗗寲鐨勮繃绋嬨
