请问两个随机变量XY不独立，他们的联合概率密度f(x,y)，怎么求E(XY)? 设二维随机变量(X,Y)联合概率密度密度如图，求E(X) E...
\u8bf7\u95ee\u4e24\u4e2a\u968f\u673a\u53d8\u91cfXY\u4e0d\u72ec\u7acb\uff0c\u4ed6\u4eec\u7684\u534f\u65b9\u5deecov(X,Y)\u5df2\u77e5\uff0c\u8bf7\u95ee\u600e\u4e48\u8ba1\u7b97\u4e24\u8005\u4e58\u79ef\u7684\u671f\u671bE(XY)?\u5229\u7528\u534f\u65b9\u5dee\u7684\u516c\u5f0f\u554aCOV(X,Y)=E[(X-E(X))(Y-E(Y))]=EXY-EX*EY
\u90a3\u4e48EXY=COV(X,Y)+EX*EYEX,EY,COV(X,Y)\u90fd\u5df2\u77e5\uff0c\u5c31\u53ef\u4ee5\u7b97\u51fa\u6765\u4e86\u3002
\u5982\u679cX\u4e0eY\u662f\u7edf\u8ba1\u72ec\u7acb\u7684\uff0c\u90a3\u4e48\u4e8c\u8005\u4e4b\u95f4\u7684\u534f\u65b9\u5dee\u5c31\u662f0\uff0c\u56e0\u4e3a\u4e24\u4e2a\u72ec\u7acb\u7684\u968f\u673a\u53d8\u91cf\u6ee1\u8db3E[XY]=E[X]E[Y]\u3002
\u4f46\u662f\uff0c\u53cd\u8fc7\u6765\u5e76\u4e0d\u6210\u7acb\u3002\u5373\u5982\u679cX\u4e0eY\u7684\u534f\u65b9\u5dee\u4e3a0\uff0c\u4e8c\u8005\u5e76\u4e0d\u4e00\u5b9a\u662f\u7edf\u8ba1\u72ec\u7acb\u7684\u3002
\u534f\u65b9\u5deeCov(X,Y)\u7684\u5ea6\u91cf\u5355\u4f4d\u662fX\u7684\u534f\u65b9\u5dee\u4e58\u4ee5Y\u7684\u534f\u65b9\u5dee\u3002\u800c\u53d6\u51b3\u4e8e\u534f\u65b9\u5dee\u7684\u76f8\u5173\u6027\uff0c\u662f\u4e00\u4e2a\u8861\u91cf\u7ebf\u6027\u72ec\u7acb\u7684\u65e0\u91cf\u7eb2\u7684\u6570\u3002
\u534f\u65b9\u5dee\u4e3a0\u7684\u4e24\u4e2a\u968f\u673a\u53d8\u91cf\u79f0\u4e3a\u662f\u4e0d\u76f8\u5173\u7684\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u82e5\u4e24\u4e2a\u968f\u673a\u53d8\u91cfX\u548cY\u76f8\u4e92\u72ec\u7acb\uff0c\u5219E[(X-E(X))(Y-E(Y))]=0\uff0c\u56e0\u800c\u82e5\u4e0a\u8ff0\u6570\u5b66\u671f\u671b\u4e0d\u4e3a\u96f6\uff0c\u5219X\u548cY\u5fc5\u4e0d\u662f\u76f8\u4e92\u72ec\u7acb\u7684\uff0c\u4ea6\u5373\u5b83\u4eec\u4e4b\u95f4\u5b58\u5728\u7740\u4e00\u5b9a\u7684\u5173\u7cfb\u3002
\u534f\u65b9\u5dee\u4e0e\u65b9\u5dee\u4e4b\u95f4\u6709\u5982\u4e0b\u5173\u7cfb\uff1a
D(X+Y)=D(X)+D(Y)+2Cov(X\uff0cY)
D(X-Y)=D(X)+D(Y)-2Cov(X\uff0cY)
\u534f\u65b9\u5dee\u4e0e\u671f\u671b\u503c\u6709\u5982\u4e0b\u5173\u7cfb\uff1a
Cov(X\uff0cY)=E(XY)-E(X)E(Y)\u3002
\u534f\u65b9\u5dee\u7684\u6027\u8d28\uff1a
\uff081\uff09Cov(X\uff0cY)=Cov(Y\uff0cX)\uff1b
\uff082\uff09Cov(aX\uff0cbY)=abCov(X\uff0cY)\uff0c\uff08a\uff0cb\u662f\u5e38\u6570\uff09\uff1b
\uff083\uff09Cov(X1+X2\uff0cY)=Cov(X1\uff0cY)+Cov(X2\uff0cY)\u3002
\u7531\u534f\u65b9\u5dee\u5b9a\u4e49\uff0c\u53ef\u4ee5\u770b\u51faCov(X\uff0cX)=D(X)\uff0cCov(Y\uff0cY)=D(Y)\u3002
\u67d0\u57ce\u5e02\u670910\u4e07\u4e2a\u5bb6\u5ead\uff0c\u6ca1\u6709\u5b69\u5b50\u7684\u5bb6\u5ead\u67091000\u4e2a\uff0c\u6709\u4e00\u4e2a\u5b69\u5b50\u7684\u5bb6\u5ead\u67099\u4e07\u4e2a\uff0c\u6709\u4e24\u4e2a\u5b69\u5b50\u7684\u5bb6\u5ead\u67096000\u4e2a\uff0c\u67093\u4e2a\u5b69\u5b50\u7684\u5bb6\u5ead\u67093000\u4e2a\u3002
\u5219\u6b64\u57ce\u5e02\u4e2d\u4efb\u4e00\u4e2a\u5bb6\u5ead\u4e2d\u5b69\u5b50\u7684\u6570\u76ee\u662f\u4e00\u4e2a\u968f\u673a\u53d8\u91cf\uff0c\u8bb0\u4e3aX\u3002\u5b83\u53ef\u53d6\u503c0\uff0c1\uff0c2\uff0c3\u3002
\u5176\u4e2d\uff0cX\u53d60\u7684\u6982\u7387\u4e3a0.01\uff0c\u53d61\u7684\u6982\u7387\u4e3a0.9\uff0c\u53d62\u7684\u6982\u7387\u4e3a0.06\uff0c\u53d63\u7684\u6982\u7387\u4e3a0.03\u3002
\u5219\uff0c\u5b83\u7684\u6570\u5b66\u671f\u671b \uff0c\u5373\u6b64\u57ce\u5e02\u4e00\u4e2a\u5bb6\u5ead\u5e73\u5747\u6709\u5c0f\u5b691.11\u4e2a\uff0c\u5f53\u7136\u4eba\u4e0d\u53ef\u80fd\u75281.11\u4e2a\u6765\u7b97\uff0c\u7ea6\u7b49\u4e8e2\u4e2a\u3002
\u8bbeY\u662f\u968f\u673a\u53d8\u91cfX\u7684\u51fd\u6570\uff1a \uff08 \u662f\u8fde\u7eed\u51fd\u6570\uff09
\u5b83\u7684\u5206\u5e03\u5f8b\u4e3a
\u82e5 \u7edd\u5bf9\u6536\u655b\uff0c\u5219\u6709:
\u2460\u5148\u6c42\u51faX\u3001Y\u7684\u8fb9\u7f18\u5206\u5e03\u5bc6\u5ea6\u51fd\u6570\u3002\u6839\u636e\u5b9a\u4e49\uff0cX\u7684\u8fb9\u7f18\u5206\u5e03\u5bc6\u5ea6\u51fd\u6570fX(x)=\u222b(0,2)f(x,y)dy=2x\u3002\u540c\u7406\uff0cY\u7684\u8fb9\u7f18\u5206\u5e03\u5bc6\u5ea6\u51fd\u6570fY(y)=\u222b(0,1)f(x,y)dx=y/2\u3002
\u2461\u6c42\u671f\u671b\u503c\u3002\u6309\u7167\u5b9a\u4e49\uff0cE(X)=\u222b(0,1)xfX(x)dx=\u222b(0,1)2x²dx=2/3\u3002\u540c\u7406\uff0cE(Y)=\u222b(0,2)yfY(y)dy=\u222b(0,2)y²dy/2=4/3\u3002E(XY)=\u222b\u222bDf(x,y)xydxdy=\u222b(0,1)x²dx\u222b(0,2)y²dy=8/9\uff0c
\u2234E(XY+1)=E(XY)+1=8/9+1=17/9\u3002
\u542b\u4e49
\u5219X\u4e3a\u8fde\u7eed\u578b\u968f\u673a\u53d8\u91cf\uff0c\u79f0f(x)\u4e3aX\u7684\u6982\u7387\u5bc6\u5ea6\u51fd\u6570\uff0c\u7b80\u79f0\u4e3a\u6982\u7387\u5bc6\u5ea6\u3002
\u5355\u7eaf\u7684\u8bb2\u6982\u7387\u5bc6\u5ea6\u6ca1\u6709\u5b9e\u9645\u7684\u610f\u4e49\uff0c\u5b83\u5fc5\u987b\u6709\u786e\u5b9a\u7684\u6709\u754c\u533a\u95f4\u4e3a\u524d\u63d0\u3002\u53ef\u4ee5\u628a\u6982\u7387\u5bc6\u5ea6\u770b\u6210\u662f\u7eb5\u5750\u6807\uff0c\u533a\u95f4\u770b\u6210\u662f\u6a2a\u5750\u6807\uff0c\u6982\u7387\u5bc6\u5ea6\u5bf9\u533a\u95f4\u7684\u79ef\u5206\u5c31\u662f\u9762\u79ef\uff0c\u800c\u8fd9\u4e2a\u9762\u79ef\u5c31\u662f\u4e8b\u4ef6\u5728\u8fd9\u4e2a\u533a\u95f4\u53d1\u751f\u7684\u6982\u7387\uff0c\u6240\u6709\u9762\u79ef\u7684\u548c\u4e3a1\u3002\u6240\u4ee5\u5355\u72ec\u5206\u6790\u4e00\u4e2a\u70b9\u7684\u6982\u7387\u5bc6\u5ea6\u662f\u6ca1\u6709\u4efb\u4f55\u610f\u4e49\u7684\uff0c\u5b83\u5fc5\u987b\u8981\u6709\u533a\u95f4\u4f5c\u4e3a\u53c2\u8003\u548c\u5bf9\u6bd4\u3002
int是什么?
绛旓細XY涓鐩稿叧鏃讹紝X銆乊涓嶄竴瀹鐙珛銆傝В锛歑銆乊涓嶇浉鍏虫槸鎸嘪銆乊鏃犵嚎鎬у叧绯伙紱X銆乊鐙珛鍒欐槸璇存槑X涓嶻鏃犱换浣曞叧绯汇闅忔満鍙橀噺鏄寚闅忔満浜嬩欢鐨勬暟閲忚〃鐜般備緥濡備竴鎵规敞鍏ユ煇绉嶆瘨鐗╃殑鍔ㄧ墿锛屽湪涓瀹氭椂闂村唴姝讳骸鐨勫彧鏁帮紱鏌愬湴鑻ュ共鍚嶇敺鎬у仴搴锋垚浜轰腑锛屾瘡浜鸿绾㈣泲鐧介噺鐨勬祴瀹氬硷紱绛夌瓑銆傚彟鏈変竴浜涚幇璞″苟涓嶇洿鎺ヨ〃鐜颁负鏁伴噺锛屼緥濡備汉鍙ｇ殑...
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绛旓細娣卞叆鎺㈣锛氫负浠涔堜笉鐩稿叧鐨闅忔満鍙橀噺X鍜孻骞朵笉鎰忓懗鐫鐙珛锛熺‘瀹烇紝鐞嗚В杩欎釜闂鐨勫叧閿湪浜庢鐜囧垎甯冪殑寰涔嬪銆傝鎴戜滑閫氳繃涓涓敓鍔ㄧ殑渚嬪瓙鏉ユ彮绀鸿繖涓蹇电殑鍐呭湪閫昏緫銆傛兂璞涓や釜闅忔満鍙橀噺(X, Y)锛屽畠浠殑鑱斿悎鍒嗗竷鎻忕粯鍦ㄤ竴涓濂囩殑鍦烘櫙涓細(X,Y)鍧囧寑鍦板垎甯冨湪鍗曚綅鍦唜² + y² = 1鐨勬瘡涓涓偣涓娿...
绛旓細鐢变簬涓嶇嫭绔嬶紝鎵浠ュ繀椤荤煡閬撹仈鍚堝瘑搴︽墠鑳芥眰銆
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绛旓細璁＄畻鍏紡涓篍(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︺備簩缁撮殢鏈哄彉閲( X锛孻)鐨勬ц川涓嶄粎涓嶺 銆乊 鏈夊叧,鑰屼笖杩樹緷璧栦簬杩涓や釜闅忔満鍙橀噺鐨勭浉浜掑叧绯汇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嗭紙X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XY涓鐩稿叧鏃讹紝X銆乊涓嶄竴瀹鐙珛銆傝В锛歑銆乊涓嶇浉鍏虫槸鎸嘪銆乊鏃犵嚎鎬у叧绯伙紱X銆乊鐙珛鍒欐槸璇存槑X涓嶻鏃犱换浣曞叧绯汇闅忔満鍙橀噺鏄寚闅忔満浜嬩欢鐨勬暟閲忚〃鐜般備緥濡備竴鎵规敞鍏ユ煇绉嶆瘨鐗╃殑鍔ㄧ墿锛屽湪涓瀹氭椂闂村唴姝讳骸鐨勫彧鏁帮紱鏌愬湴鑻ュ共鍚嶇敺鎬у仴搴锋垚浜轰腑锛屾瘡浜鸿绾㈣泲鐧介噺鐨勬祴瀹氬硷紱绛夌瓑銆傚彟鏈変竴浜涚幇璞″苟涓嶇洿鎺ヨ〃鐜颁负鏁伴噺锛屼緥濡備汉鍙ｇ殑...
绛旓細P(X,Y) * 81/2 = 1 鎵浠 P(X,Y) = 2/81 (x>0, y>0, x+y<9)涓夎褰鐨勯潪0閮ㄥ垎灏辨槸涓夎褰㈠幓闄3杈圭殑閮ㄥ垎浜嗐(2) 鍥犱负x+y<9,鎵浠鐨勫彇鍊,涓瀹氱▼搴︿笂渚濊禆x ,鍥犳XY涓や釜闅忔満鍙橀噺骞涓嶇嫭绔 銆(3) 鍥犱负f(x,y)鎭掍负2/81锛屽疄闄呬笂f(x,y)灏辨槸姒傜巼瀵嗗害鍑芥暟锛屽湪澶氱淮閲孭(X,Y)...
绛旓細XY涓鐩稿叧鏃讹紝X銆乊涓嶄竴瀹鐙珛銆傝В锛歑銆乊涓嶇浉鍏虫槸鎸嘪銆乊鏃犵嚎鎬у叧绯伙紱X銆乊鐙珛鍒欐槸璇存槑X涓嶻鏃犱换浣曞叧绯汇闅忔満鍙橀噺鏄寚闅忔満浜嬩欢鐨勬暟閲忚〃鐜般備緥濡備竴鎵规敞鍏ユ煇绉嶆瘨鐗╃殑鍔ㄧ墿锛屽湪涓瀹氭椂闂村唴姝讳骸鐨勫彧鏁帮紱鏌愬湴鑻ュ共鍚嶇敺鎬у仴搴锋垚浜轰腑锛屾瘡浜鸿绾㈣泲鐧介噺鐨勬祴瀹氬硷紱绛夌瓑銆傚彟鏈変竴浜涚幇璞″苟涓嶇洿鎺ヨ〃鐜颁负鏁伴噺锛屼緥濡備汉鍙ｇ殑...
