设随机变量X与Y相互独立，且X~N(2，1),Y~N(-2，4),Z=3X-2Y+4，求：D(Z) 与 P{Z<=9} 设随机变量X和Y相互独立,且X~N(3,4),Y~(2,9)...
\u8bbe\u968f\u673a\u53d8\u91cfX\u548cY\u76f8\u4e92\u72ec\u7acb\uff0c\u4e14X~N\uff082,3\uff09\uff0cY~N\uff081,4\uff09\uff0c\u52193X-4Y\u7684\u5206\u5e03\u4e3ay\u5e94\u662f\u670d\u4ecey~n(1,4)\u5427
\u7531\u5df2\u77e5\u5f97ex=0\uff0cdx=1\uff0cey=1\uff0cdy=4\uff0c\u4e8e\u662fe(x-y)=ex-ey=-1\uff0c
x,y\u76f8\u4e92\u72ec\u7acb\uff0c\u6240\u4ee5d(x-y)=dx+d(-y)=dx+dy=5\u3002
\u6545x-y~n(-1\uff0c5)
3x-y\u8fd8\u662f\u6b63\u6001\u5206\u5e03\uff0c\u5229\u7528\u516c\u5f0f e(ax+by)=+ae(x)+be(y)\uff0cd(ax+by)=+a²d(x)+b²d(y)\u3002
\u968f\u673a\u53d8\u91cf\u76f8\u4e92\u72ec\u7acb\u53ef\u4ee5\u63a8\u51fa\u7ebf\u6027\u4e0d\u76f8\u5173\uff0c\u800c\u7ebf\u6027\u4e0d\u76f8\u5173\u4e0d\u80fd\u63a8\u51fa\u968f\u673a\u53d8\u91cf\u76f8\u4e92\u72ec\u7acb\uff0c\u6240\u4ee5\u968f\u673a\u53d8\u91cf\u7ebf\u6027\u4e0d\u76f8\u5173\u662f\u76f8\u4e92\u72ec\u7acb\u7684\u5fc5\u8981\u4e0d\u5145\u5206\u6761\u4ef6\u3002
\u5982\u679c\u8868\u793a\u8bd5\u9a8c\u7ed3\u679c\u7684\u53d8\u91cfx\uff0c\u5176\u53ef\u80fd\u53d6\u503c\u4e3a\u67d0\u8303\u56f4\u5185\u7684\u4efb\u4f55\u6570\u503c\uff0c\u4e14x\u5728\u5176\u53d6\u503c\u8303\u56f4\u5185\u7684\u4efb\u4e00\u533a\u95f4\u4e2d\u53d6\u503c\u65f6\uff0c\u5176\u6982\u7387\u662f\u786e\u5b9a\u7684\uff0c\u5219\u79f0x\u4e3a\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\u6574\u7406\u3002\u5f15\u5165\u968f\u673a\u53d8\u91cf\u7684\u6982\u5ff5\u540e\uff0c\u5bf9\u968f\u673a\u8bd5\u9a8c\u7684\u6982\u7387\u5206\u5e03\u7684\u7814\u7a76\u5c31\u8f6c\u4e3a\u5bf9\u968f\u673a\u53d8\u91cf\u6982\u7387\u5206\u5e03\u7684\u7814\u7a76\u4e86\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u6ce8\u610f\u4e8b\u9879\uff1a
\u968f\u673a\u53d8\u91cf\u7b49\u4e8e\u67d0\u4e2a\u503c\u6216\u8005\u5904\u4e8e\u67d0\u4e2a\u533a\u95f4\u662f\u4e00\u4e2a\u4e8b\u4ef6\u3002\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\u72ec\u7acb\u8bf4\u7684\u662f\u4e00\u7ec4\u4e8b\u4ef6\u4e0e\u53e6\u4e00\u7ec4\u72ec\u7acb\u3002
\u968f\u673a\u53d8\u91cf\u662f\u5b9a\u4e49\u5728\u6837\u672c\u7a7a\u95f4\u4e0a\u7684\u5b9e\u503c\u51fd\u6570\uff0c\u662f\u628a\u968f\u673a\u73b0\u8c61\u7684\u7ed3\u679c\u6570\u91cf\u5316\uff0c\u4fbf\u4e8e\u8fdb\u884c\u6570\u5b66\u5904\u7406\u3002\u6982\u7387\u8bba\u4e2d\u7684\u968f\u673a\u53d8\u91cf\u662f\u4e00\u79cd\u968f\u673a\u53d6\u503c\u7684\u53d8\u91cf\u4e14\u4f34\u968f\u4e00\u4e2a\u5206\u5e03\u3002
\u4e8b\u4ef6\u7684\u76f8\u4e92\u72ec\u7acb\u53ef\u5b9a\u4e49\u8bd5\u9a8c\u7684\u76f8\u4e92\u72ec\u7acb\uff0c\u8bd5\u9a8c\u7684\u76f8\u4e92\u72ec\u7acb\u53ef\u63a8\u51fa\u4e00\u4e9b\u4e8b\u4ef6\u7684\u76f8\u4e92\u72ec\u7acb\u3002\u8bd5\u9a8c\u7684\u72ec\u7acb\u6027\u548c\u968f\u673a\u53d8\u91cf\u7684\u72ec\u7acb\u6027\u90fd\u662f\u5728\u4e8b\u4ef6\u72ec\u7acb\u6027\u7684\u57fa\u7840\u4e0a\u6765\u5b9a\u4e49\u7684\uff0c\u968f\u673a\u53d8\u91cf\u53d6\u67d0\u4e2a\u503c\u6216\u53d6\u67d0\u4e2a\u8fde\u7eed\u533a\u95f4\u65f6\uff0c\u5c31\u662f\u8868\u793a\u67d0\u4e8b\u4ef6\u3002
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u72ec\u7acb\u540c\u5206\u5e03\u53d8\u91cf
\u53c2\u8003\u8d44\u6599\u6765\u6e90\uff1a\u767e\u5ea6\u767e\u79d1-\u968f\u673a\u53d8\u91cf
E(aX+bY)=aE(X)+bE(Y)
D(aX+bY)=a²D(X)+b²D(Y)
由已知得E(X) = 2, D(X) = 1,E(Y) = -2, D(Y) = 4,则：
E(Z) = E(3X-2Y+4) = 3E(X) - 2E(Y) + E(4) = 14
D(Z) = D(3X-2Y+4) = 9D(X) + 4D(Y) + 0 = 25
所以 Z~N(14,25)
P{Z<=9} = P{(z-14)/5 <= (9-14)/5} = φ(-1) = 1-φ(1) = 0.16
Z=3X-2Y+4
E(Z)=E(3X-2Y+4)
=E(3X)-E(2Y)+E(4)
=3*2-2*2+4
=9
D(Z)=D(3X-2Y+4)
=D(3X)+D(2Y)+D(4)
=9*1+4*4
=25
P{Z<=9}
=P0.5
绛旓細鍒╃敤鎺ㄥ箍鐨勫嵎绉叕寮忥細鍏朵腑z=g(x,y)锛岄偅涔坹=h(x,z),fz(z)=鈭玣(x,h(x,z))脳|h瀵箊鐨勫亸瀵兼暟|dx濂楀湪棰樹腑X锛孻鐩镐簰鐙珛涓擸=XZ锛屽甫鍏ュ叕寮忓嵆鍙傞殢鏈鸿瘯楠屽悇绉嶇粨鏋滅殑瀹炲煎崟鍊煎嚱鏁般傞殢鏈轰簨浠朵笉璁轰笌鏁伴噺鏄惁鐩存帴鏈夊叧锛岄兘鍙互鏁伴噺鍖栵紝鍗抽兘鑳界敤鏁伴噺鍖栫殑鏂瑰紡琛ㄨ揪銆傚惈涔 鍒橷涓鸿繛缁瀷闅忔満鍙橀噺锛绉癴(...
绛旓細/12=1/3锛孍(X^2)=D(X)+E(X)^2=4/3锛屽洜涓篩~N(2,4)锛屾墍浠(Y)=2,D(Y)=4锛孍(Y^2)=D(Y)+E(Y)^2=8锛屾牴鎹湡鏈涗笌鏂瑰樊鐨勬ц川鍙緱锛欵(XY)=E(X)E(Y)=2锛孍((XY)^2)=E((X^2)(Y^2))=E(X^2)E(Y^2)=32/3锛孌(XY)=E((XY)^2)-[E(XY)]^2=20/3銆
绛旓細璁颁綇姝ｆ佸垎甯冪殑鍩烘湰鍏紡 濡傛灉涓や釜鐩镐簰鐙珛鐨勬鎬佸垎甯 X~N(渭1, m²)锛孻~N(渭2锛宯²)閭ｄ箞Z=aX卤bY浠嶇劧鏈嶄粠姝ｆ佸垎甯冿紝Z~N(a渭1卤b渭2锛宎²m²+b²n²)浜庢槸杩欓噷鐨2X+3Y~N(2*1+3*2锛4*2+9*2)鍗虫湇浠庡垎甯僋(8锛26)
绛旓細2016-07-05 璁鹃殢鏈哄彉閲廥涓嶻鐩镐簰鐙珛,涓旈兘鏈嶄粠N(0,1/2),鍒橢(|... 147 2018-08-16 璁鹃殢鏈哄彉閲弜y鐩镐簰鐙珛,x~n(1,2),y~n(0,1),... 4 2012-11-08 璁鹃殢鏈哄彉閲廥涓嶻鐩镐簰鐙珛,涓擷~U(0,1),Y~e(1),... 94 2013-07-21 璁鹃殢鏈哄彉閲廥涓嶻鐩镐簰鐙珛,涓擷~N(0,1),Y~N(1,1... 25 ...
绛旓細浣犲ソ锛佸彲浠ュ鍥惧埄鐢ㄦ硦鏉惧垎甯冨叕甯冭绠楋紝绛旀鏄18e^(-6)銆傜粡娴庢暟瀛﹀洟闃熷府浣犺В绛旓紝璇峰強鏃堕噰绾炽傝阿璋紒
绛旓細D(X)=1/0.5^2=4,D(Y)=36/12=3 浜庢槸D(X-2Y)=D(X)+4D(Y)=4+4X3=16.
绛旓細銆愮瓟妗堛戯細f(x,y)=(1/锛4蟺锛)*e^[-x^2/2-(y-1)^2/8]F(x,y)=FX(x)*FY(y),F(0,1)=FX(0)*FY(1)=0.5*0.5=0.25
绛旓細闅忔満鍙橀噺X涓嶻鐩镐簰鐙珛锛屼笖D(X)=1锛孌(Y)=2 鍒橠(2X-3Y)=2^2D(X)+3^2D(Y)=4x1+9x2 =4+18 =22 鐢变簬闅忔満鍙橀噺X鐨鍙栧 鍙彇鍐充簬姒傜巼瀵嗗害鍑芥暟鐨勭Н鍒嗭紝鎵浠ユ鐜囧瘑搴﹀嚱鏁板湪涓埆鐐逛笂鐨勫彇鍊煎苟涓嶄細褰卞搷闅忔満鍙橀噺鐨勮〃鐜般傚鏋滀竴涓嚱鏁板拰X鐨勬鐜囧瘑搴﹀嚱鏁板彇鍊间笉鍚岀殑鐐瑰彧鏈夋湁闄愪釜銆佸彲鏁版棤闄愪釜鎴栬呯浉瀵...
绛旓細绛旀鏄疊锛屽彲浠ュ弽鐢ㄦ柟宸绠楀叕寮忔潵璁＄畻骞虫柟鐨勬湡鏈涖傝閲囩撼锛岃阿璋紒
绛旓細鏍规嵁棰樼洰涓殑璁惧畾锛屾垜浠湁闅忔満鍙橀噺 X 鏈嶄粠鍙傛暟涓 4 鐨勬硦鏉惧垎甯 P(4)锛岃岄殢鏈哄彉閲 Y 鏈嶄粠鍙傛暟涓 (8, 0.5) 鐨勪簩椤瑰垎甯 B(8, 0.5)銆傛垜浠渶瑕佹眰瑙 Z = X - 2Y + 10 鐨勬湡鏈涘 E(Z) 鍜屾柟宸 V(Z)銆傞鍏堬紝鎴戜滑鏉ヨ绠 Z 鐨勬湡鏈涘 E(Z)銆傜敱浜 X 鍜 Y 鏄鐩镐簰鐙珛鐨勯殢鏈哄彉閲忥紝鎴戜滑...
