二维随机变量的E(X)和E(Y)也已经求出，求COV(X,Y)，公式COV(XY)=E(XY)-E(X)E(Y)中的E(XY)是怎么得出的？
\u5173\u4e8e\u4e8c\u5143\u79bb\u6563\u578b\u968f\u673a\u53d8\u91cf\u7684\u534f\u65b9\u5dee\u7684\u8ba1\u7b97\u516c\u5f0fCov(X,Y)=E(XY)-E(X)E(Y)\u4e2d\uff0cE(EY)\u662f\u600e\u4e48\u7b97\u51fa\u6765\u5462\uff1f1\uff09\u5982\u679cXY\u72ec\u7acb
E(XY)=E(X)E(Y\uff09
2\uff09\u5982\u679c\u4e0d\u72ec\u7acb\uff0c\u82e5\u662f\u79bb\u6563\u7684\uff0c\u5219
\u2211\u2211XiYjPij
(i=1,2,3\u2026..,j=1,2,3\u2026..)
\u82e5\u662f\u8fde\u7eed\u7684\uff0c\u5219\u222b\u222bxyf(xy)dxdy
(f(xy)\u4e3a\u5bc6\u5ea6\u51fd\u6570)
\u6c57S\u8fd9\u91cc\u771f\u4e0d\u597d\u6253\u51fa\u6765\u79ef\u5206\u4e0a\u4e0b\u9650\uff0c\u5b9a\u4e49\u662f\u4ece\u8d1f\u65e0\u7a77\u79ef\u5230\u6b63\u65e0\u7a77\uff0c\u4f46\u5b9e\u9645\u95ee\u9898\u662f\u4ece\u5bc6\u5ea6\u51fd\u6570\u4e0d\u4e3a\u96f6\u7684\u8303\u56f4\u79ef\u5206\uff0c\u79bb\u6563\u7684\u4e0d\u7528\u8bf4\u4e86\u5427\uff0c\u5c31\u662f\u628a\u5b83\u4eec\u7684\u6570\u503c\u4e58\u4ee5\u8054\u5408\u6982\u7387\u518d\u76f8\u52a0
E[(X-E(X))(Y-E(Y))]=E[XY-XE(Y)-E(X)Y+E(X)E(Y)]
=E(XY)-E(X)E(Y)-E(X)E(Y)+E(X)E(Y)
=E(XY)-E(X)E(Y)
如果是连续性随即变量则求出积分区间为负无穷至正无穷的∫xyf(xy)就可以了
绛旓細璁＄畻鍏紡涓E(XY)=鈭埆xyf(x锛寉)dxdy锛岀Н鍒嗚寖鍥存槸鏁翠釜骞抽潰锛屽叾涓璮(x,y)鏄仈鍚堟鐜囧瘑搴︺浜岀淮闅忔満鍙橀噺( X锛Y)鐨勬ц川涓嶄粎涓嶺 銆乊 鏈夊叧,鑰屼笖杩樹緷璧栦簬杩欎袱涓闅忔満鍙橀噺鐨鐩镐簰鍏崇郴銆傚洜姝わ紝閫愪釜鍦版潵鐮旂┒X鎴朰鐨勬ц川鏄笉澶熺殑锛岃繕闇灏锛圶锛孻锛変綔涓轰竴涓暣浣撴潵鐮旂┒銆傝E鏄竴涓殢鏈鸿瘯楠岋紝瀹冪殑鏍锋湰绌洪棿鏄...
绛旓細锛3锛塃(XY)=绉垎X(0鍒1) 绉垎Y(0鍒1-x) xy*p(x,y)dydx=绉垎X(0鍒1) x(1-x)^2 dx=(1/2*x^2-2/3*x^3+1/4*x^4)|(0,1)=1/12銆傦紙4锛夋墍浠ワ紝鏍规嵁瀹氫箟锛欳OV(X,Y)=E(XY)-E(X)*E(Y)=1/12-1/3*1/3= -1/36銆傁XY=COV(X,Y)/(鏍瑰彿D(X)*鏍瑰彿D(Y))=-1...
绛旓細f(x,y)~(u1,u2,蟽1²,蟽2²,蟻)鍏朵腑u1=0锛寀2=0锛屜1²=16锛屜2²=25锛屜=Cov(x,y)=12 鎶婃暟瀛椾唬鍏ュ嵆鍙
绛旓細涓嶆槸銆俋 Y鐩镐簰鐙珛鍙互鎺ㄥ嚭E(XY)=E(X)E(Y) 锛屼絾瀹冪殑閫嗗懡棰樹笉鎴愮珛銆備笉鐩稿叧鍜屼笉鐙珛涓嶇瓑浠凤紝鍙湁鏌愪簺鏃跺欎笉鐩稿叧鍜屼笉鐙珛鏄瓑浠风殑銆佹瘮濡傝浜岀淮姝ｆ闅忔満鍙橀噺銆傝嫢P(A)>0锛孭(B)>0鍒橝锛孊鐩镐簰鐙珛涓嶢锛孊浜掍笉鐩稿涓嶈兘鍚屾椂鎴愮珛锛屽嵆鐙珛蹇呯浉瀹癸紝浜掓枼蹇呰仈绯汇傚鏄撴帹骞匡紝璁続锛孊锛孋鏄笁涓簨浠讹紝濡傛灉...
绛旓細E(X)E(Y) 鍒欒〃绀洪殢鏈哄彉閲 X 鍜 Y 鐨勭Н鐨勬湡鏈涘笺傝繖涔熷氨鏄锛孍(X)E(Y) 鏄皢涓や釜闅忔満鍙橀噺鐨姒傜巼鍒嗗竷鐩镐箻鍚庯紝姹傚嚭鎵鏈変箻绉殑姒傜巼鍒嗗竷鐨勬湡鏈涘笺傝繖鍙互鐢ㄤ簬璁＄畻涓や釜闅忔満鍙橀噺涔嬮棿瀛樺湪鏌愮渚濊禆鍏崇郴鏃剁殑鏈熸湜缁撴灉锛屼緥濡傚湪淇濋櫓琛屼笟涓绠楄禂娆炬敮浠樼殑鏈熸湜閲戦銆傛荤殑鏉ヨ锛孍(X) 路 E(Y) 鍜 E(X)E...
绛旓細E(X+Y)=E(X)+E(Y)=瀵箈f(x)鍦ㄥ搴斿畾涔夊煙涓婄殑绉垎+瀵箉f(y)鍦ㄥ搴斿畾涔夊煙涓婄殑绉垎 E(XY)=瀵xy*f(x,y)鍦ㄥ搴斿畾涔夊煙涓婄殑绉垎
绛旓細XY鐙珛锛岄偅涔圗锛圶Y锛夛紳E锛圶锛塃锛圷锛夛紝浜庢槸baiCOV锛圶Y锛夛紳E锛伙紙X锛岴锛圶锛夛級锛圷锛E锛圷锛锛夛冀锛滶锛圶Y锛夛紞E锛圶锛塃锛圷锛夛紳0銆傝嚦浜庝负浠涔圶Y鐙珛E锛圶Y锛夛紳E锛圶锛塃锛圷锛夛紝杩欐槸鍥犱负XY鐨勪袱涓垎甯僷xy锛坸y锛夛紳px锛坸锛塸y锛坹锛夈傚崗鏂瑰樊鏄袱涓鍙橀噺鐨鎬讳綋璇樊锛屽畠涓嶅悓浜庝竴涓彉閲忚宸殑鏂瑰樊銆
绛旓細鍚岀悊锛孍(X²)=鈭(0,1)x²[e^(-x)-e^(-2x)]dx=鈥=7/4-5/e+(5/4)/e²銆傗埓D(X)=E(X²)-[E(X)]²=7/4-5/e+(5/4)/e²-[(3/4)(1+1/e²)-2/e]²銆侲(XY)=∬Dxyf(x,y)dxdy=鈭(0,1)x²鈭(0,1)...
绛旓細E(XY)=-2路4路1/4+(-1)路1路1/4+1路1路1/4+2路1路1/4=0 鈭礐ov(X锛孻)=E(XY)-E(X)路E(Y)=0 鈭碭涓嶻涓嶇浉鍏炽(2)P(X=-2锛孻=1)=0鈮燩(X=-2)路P(Y=1)鈭碭涓嶻涓嶇浉浜掔嫭绔嬨傛牴鎹闅忔満鍙橀噺鐨涓嶅悓锛岃仈鍚堟鐜囧垎甯冪殑琛ㄧず褰㈠紡涔熶笉鍚屻傚浜庣鏁ｅ瀷闅忔満鍙橀噺锛岃仈鍚堟鐜囧垎甯冨彲浠ヤ互...
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