设随机变量X，Y相互独立，且X~N（1,5），Y~N（2，3），试求Z=2X-3Y+1的概率密度,求。。。。 若随机变量X~N(-2，4)，Y~N(3，9)，且X与Y相互...
\u8bbe\u968f\u673a\u53d8\u91cfX\u4e0eY\u76f8\u4e92\u72ec\u7acb\uff0cX~N\uff082,1),Y~N(1,2),\u5219Z=2X-Y+3\u7684\u5bc6\u5ea6\u51fd\u6570\u8868\u8fbe\u5f0f\u4e3aE(Z)=? ,D(Z)=? ,f\uff08Z)=?\uff081\uff09\u56e0\u4e3aX~N\uff082,1),Y~N(1,2),\u6545;EX=2,DX=1,EY=1,DY=2
\u518d\u7531\u671f\u671b\u4e0e\u65b9\u5dee\u7684\u6027\u8d28\uff1a
E(Z)=E(2X-Y+3)=2E(X)-E(Y)+3=2*2-1+3=6
D(Z)=D(2X-Y+3)=2^2D(X)+\uff08-1)^2D(Y)=4*1+2=6
\u53c8\u56e0\u4e3a\u72ec\u7acb\u7684\u6b63\u6001\u5206\u5e03\u7684\u7ebf\u6027\u51fd\u6570\u8fd8\u662f\u6b63\u6001\u5206\u5e03\uff0c\u6545\uff1aZ~N(6,6)\uff0cf(z)\u53ef\u6839\u636e\u6b63\u6001\u5206\u5e03\u7684\u516c\u5f0f\u5199\u51fa
\uff082\uff09\u7531\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u5206\u5e03\u5217\u7684\u6027\u8d28\uff0c\u6240\u6709\u70b9\u5bf9\u5e94\u7684\u6982\u7387\u4e4b\u548c\u4e3a 1\uff0c
\u6240\u4ee5\uff1a0.1+0.3+0.1+a+0.2=1
\u7531\u6b64\u6c42\u5f97\uff1aa=0.3
\u800c 0\uff1cX\u22642\u65f6\uff0cx\u53ea\u53d61\u4e0e2\u4e24\u4e2a\u70b9\uff0c\u8fd9\u4e24\u4e2a\u70b9\u5bf9\u5e94\u7684\u6982\u7387\u5206\u522b\u662f0.1\u4e0e0.3\uff0c
\u6545\uff1aP\uff080\uff1cX\u22642\uff09=0.1+0.3=0.4
\u9996\u5148\uff0c\u6839\u636eX\u4e0eY\u662f\u76f8\u4e92\u72ec\u7acb\u7684\u6b63\u6001\u5206\u5e03\uff0c\u56e0\u6b64\u5b83\u4eec\u7684\u7ebf\u6027\u7ec4\u5408\u4e5f\u662f\u670d\u4ece\u6b63\u6001\u5206\u5e03\uff1b\u518d\u6839\u636e\u7edf\u8ba1\u91cf\u4e2d\u7684\u76f8\u5173\u5b9a\u7406\uff0c\u6c42\u51fa\u8fd9\u4e00\u5206\u5e03\u7684\u4e24\u4e2a\u53c2\u6570\u5373\u53ef\u3002
\u968f\u673a\u53d8\u91cfX\uff5eN\uff08-3\uff0c1\uff09\uff0cY\uff5eN\uff082\uff0c1\uff09\uff0c\u4e14X\u4e0eY\u76f8\u4e92\u72ec\u7acb\u2234Z=X-2Y+7\u4e5f\u670d\u4ece\u6b63\u6001\u5206\u5e03\u53c8\u7531\u4e8eEZ=E\uff08X-2Y+7\uff09=E\uff08X\uff09-2E\uff08Y\uff09+E\uff087\uff09=-3-2•2+7=0\uff0cD\uff08X-2Y+7\uff09=D\uff08X\uff09+\uff08-2\uff092D\uff08Y\uff09+D\uff087\uff09=1+4+0=5\u2234Z\uff5eN\uff080\uff0c5\uff09\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u968f\u673a\u4e8b\u4ef6\u6570\u91cf\u5316\u7684\u597d\u5904\u662f\u53ef\u4ee5\u7528\u6570\u5b66\u5206\u6790\u7684\u65b9\u6cd5\u6765\u7814\u7a76\u968f\u673a\u73b0\u8c61\u3002\u4f8b\u5982\u67d0\u4e00\u65f6\u95f4\u5185\u516c\u5171\u6c7d\u8f66\u7ad9\u7b49\u8f66\u4e58\u5ba2\u4eba\u6570\uff0c\u7535\u8bdd\u4ea4\u6362\u53f0\u5728\u4e00\u5b9a\u65f6\u95f4\u5185\u6536\u5230\u7684\u547c\u53eb\u6b21\u6570\uff0c\u706f\u6ce1\u7684\u5bff\u547d\u7b49\u7b49\uff0c\u90fd\u662f\u968f\u673a\u53d8\u91cf\u7684\u5b9e\u4f8b\u3002
\u5728\u505a\u5b9e\u9a8c\u65f6\uff0c\u5e38\u5e38\u662f\u76f8\u5bf9\u4e8e\u8bd5\u9a8c\u7ed3\u679c\u672c\u8eab\u800c\u8a00\uff0c\u6211\u4eec\u4e3b\u8981\u8fd8\u662f\u5bf9\u7ed3\u679c\u7684\u67d0\u4e9b\u51fd\u6570\u611f\u5174\u8da3\u3002\u4f8b\u5982\uff0c\u5728\u63b7\u9ab0\u5b50\u65f6\uff0c\u6211\u4eec\u5e38\u5e38\u5173\u5fc3\u7684\u662f\u4e24\u9897\u9ab0\u5b50\u7684\u70b9\u548c\u6570\uff0c\u800c\u5e76\u4e0d\u771f\u6b63\u5173\u5fc3\u5176\u5b9e\u9645\u7ed3\u679c\uff1b
\u5c31\u662f\u8bf4\uff0c\u6211\u4eec\u5173\u5fc3\u7684\u4e5f\u8bb8\u662f\u5176\u70b9\u548c\u6570\u4e3a7\uff0c\u800c\u5e76\u4e0d\u5173\u5fc3\u5176\u5b9e\u9645\u7ed3\u679c\u662f\u5426\u662f\uff081\uff0c6\uff09\u6216\uff082\uff0c5\uff09\u6216\uff083\uff0c4\uff09\u6216\uff084\uff0c3\uff09\u6216\uff085\uff0c2\uff09\u6216\uff086\uff0c1\uff09\u3002\u6211\u4eec\u5173\u6ce8\u7684\u8fd9\u4e9b\u91cf\uff0c\u6216\u8005\u66f4\u5f62\u5f0f\u7684\u8bf4\uff0c\u8fd9\u4e9b\u5b9a\u4e49\u5728\u6837\u672c\u7a7a\u95f4\u4e0a\u7684\u5b9e\u503c\u51fd\u6570\uff0c\u79f0\u4e3a\u968f\u673a\u53d8\u91cf\u3002
E(X)=1，E(Y)=2，D(X)=5，D(Y)=3
所以，Z=2X-3Y+1，服从正态分布
E(Z)=2E(X)-3E(Y)+1=-3，
D(Z)=4D(X)+9D(Y)=47
所以，Z~N(-3，47)
概率密度就好写了
绛旓細X-Y~N(-1锛5)銆傜敱宸茬煡寰桬X=0锛孌X=1锛孍Y=1锛孌Y=4锛屼簬鏄疎(X-Y)=EX-EY=-1锛X锛孻鐩镐簰鐙珛锛屾墍浠(X-Y)=DX+D(-Y)=DX+DY=5銆傛晠X-Y~N(-1锛5)
绛旓細銆愮瓟妗堛戯細E(X)=1,D(X)=1;E(Y)=-2,D(Y)=1.鍥犳E(2X+Y)=2E(X)+E(Y)=2脳1+(-2)=0,D(2X+Y)=22D(X)+D(Y)=4脳1+1=5.
绛旓細銆愮瓟妗堛戯細
绛旓細e(z)=e(2x-y)=2e(x)-e(y)=2-0=2 d(z)=d(2x-y)=4d(x)+d(y)=4(2)+1=9 鎵浠=2X-Y+3=(2,9)涓涓簩缁存鎬2113鍒嗗竷鐨勮竟缂樺垎甯冪殑鍜屾绘槸姝ｆ佸垎甯冦傜壒鍒殑涓や釜鐙珛姝ｆ佸垎甯冪殑鍜屾绘槸姝ｆ佸垎甯冦
绛旓細銆愮瓟妗堛戯細D 鐢变簬鍙橀噺X鍜孻鐩镐簰鐙珛锛屽垯 E锛圸锛夛紳E锛圶锛媃锛夛紳E锛圶锛夛紜E锛圷锛夛紳渭1锛嬑2D锛圸锛夛紳D锛圶锛媃锛夛紳D锛圶锛夛紜D锛圷锛夛紳蟽12锛嬒22鏁匷锝濶锛埼1锛嬑2锛屜12锛嬒22锛夈
绛旓細銆愮瓟妗堛戯細姝ｆ佸垎甯冪殑绾挎х粍鍚堬紟鍥犱负X锛孻鐩镐簰鐙珛锛屼笖閮芥湇浠庢鎬佸垎甯冿紝鎵浠ュ畠浠殑绾挎х粍鍚圵=2X-Y鏈嶄粠姝ｆ佸垎甯冿紝浠庤屽嚱鏁癦=W+3涔熸湇浠庢鎬佸垎甯冿紟璁＄畻鏁板瓧鐗瑰緛锛屽緱 E(Z)=2E(X)-E(Y)+3=5锛孌(Z)=4D(X)+D(Y)=17锛庝簬鏄紝姹傚緱鍑芥暟z鐨勬鐜囧瘑搴︺傛眰浜岀淮闅忔満鍙橀噺鐨勫嚱鏁扮殑鍒嗗竷锛屼竴鑸渶瑕佷粠鍒嗗竷...
绛旓細璁 X 涓 Y 鏄袱涓殢鏈哄彉閲忥紝鍒橠锛圶锛峐锛夛紳D锛圶锛夛紜D锛圷锛夛紞2Cov锛X锛孻锛夛紝鍏朵腑鍗忔柟宸瓹ov锛圶锛孻锛夛紳E锝涳蓟x锛岴锛圶锛夛冀锛粂锛岴锛圷锛夛冀锝濄傜壒鍒殑锛屽綋X锛孻鏄袱涓笉鐩稿叧鐨勯殢鏈哄彉閲忔椂锛孋ov锛圶锛孻锛=0锛屽垯D锛圶锛峐锛夛紳D锛圶锛夛紜D锛圷锛夈傚洜涓闅忔満鍙橀噺X涓Y鐩镐簰鐙珛锛屼笖Z锛漍锛峐锛屾墍浠...
绛旓細闅忔満鍙橀噺X涓Y鐩镐簰鐙珛锛E(XY)=E(X)*E(Y)=1*(1/3)=1/3 鎸囨暟鍒嗗竷涓庡垎甯冩寚鏁版棌鐨勫垎绫讳笉鍚岋紝鍚庤呮槸鍖呭惈鎸囨暟鍒嗗竷浣滀负鍏舵垚鍛樹箣涓鐨勫ぇ绫绘鐜囧垎甯冿紝涔熷寘鎷鎬佸垎甯冿紝浜岄」鍒嗗竷锛屼冀椹垎甯冿紝娉婃澗鍒嗗竷绛夌瓑銆傚鏋滀竴涓殢鏈哄彉閲忓憟鎸囨暟鍒嗗竷锛屽綋s,t>0鏃舵湁P(T>t+s|T>t)=P(T>s)銆傚嵆锛屽鏋淭鏄煇涓...
绛旓細E(X+Y) = E(X)+E(Y) = -1+1 =0 D(X+Y) = D(X)+D(Y) = 1+3 = 4 X+Y ~ N (0, 4)P(X+Y>2)=P(Z > (2-0)/2 )=P(Z >1)
绛旓細闅忔満鍙橀噺鐩镐簰鐙珛鍙互鎺ㄥ嚭绾挎т笉鐩稿叧锛岃岀嚎鎬т笉鐩稿叧涓嶈兘鎺ㄥ嚭闅忔満鍙橀噺鐩镐簰鐙珛锛鎵浠ラ殢鏈哄彉閲忕嚎鎬т笉鐩稿叧鏄浉浜掔嫭绔嬬殑蹇呰涓嶅厖鍒嗘潯浠躲傚鏋滆〃绀鸿瘯楠岀粨鏋滅殑鍙橀噺x锛鍏跺彲鑳藉彇鍊间负鏌愯寖鍥村唴鐨勪换浣曟暟鍊锛屼笖x鍦ㄥ叾鍙栧艰寖鍥村唴鐨勪换涓鍖洪棿涓彇鍊兼椂锛屽叾姒傜巼鏄‘瀹氱殑锛屽垯绉皒涓鸿繛缁瀷闅忔満鍙橀噺鏁寸悊銆傚紩鍏ラ殢鏈哄彉閲忕殑姒傚康鍚庯紝瀵...
