设二维随机变量服从圆域的均匀分布, 设二维随机变量服从圆域x^2+y^2
二维随机变量服从圆域x^2+y^2绛旓細浜岀淮闅忔満鍙橀噺鏈嶄粠鍦嗗煙x^2+y^2
绛旓細浜岀淮闅忔満鍙橀噺鏈嶄粠鍦嗗煙x^2+y^2<=R^2鐨勫潎鍖鍒嗗竷 鎵浠锛寉鐨勬鐜囧垎甯冨嚱鏁癴锛坸锛寉锛=1/S=1/锛埾R^2锛墄^2+y^2<=R^2 0 鍏朵粬 E锛圸锛=鈭珃f锛坺锛塪z=鈭(x^2+y^2)^0.5/锛埾R^2锛塪xdy=鈭玠胃(0~2蟺)鈭玶^2/(蟺R^2)dr(0~R)=2R/3 ...
绛旓細浜岀淮闅忔満鍙橀噺鏈嶄粠鍦嗗煙x^2+y^2<=R^2鐨勫潎鍖鍒嗗竷 鎵浠锛寉鐨勬鐜囧垎甯冨嚱鏁癴锛坸锛寉锛=1/S=1/锛埾R^2锛 x^2+y^2<=R^2 0 鍏朵粬 E锛圸锛=鈭珃f锛坺锛塪z=鈭(x^2+y^2)^0.5/锛埾R^2锛塪xdy=鈭玠胃(0~2蟺)鈭玶^2/(蟺R^2)dr(0~R)=2R/3 ...
绛旓細鐢诲嚭鍥惧舰锛屽x绉垎寰楀埌fY(y),鐢讳竴鏉℃按骞崇嚎浜鍦浜2鐐癸紝鍏舵í鍧愭爣鍒嗗埆鏄-鈭歊^2-y^2,鈭歊^2-y^2,涔熷氨鏄Н鍒嗕笂涓嬮檺銆傚y绉垎鍙緱鍒癴X(x).鍚岀悊鐢讳竴鏉″瀭鐩寸嚎浜ゅ渾浜2鐐癸紝绾靛潗鏍囧垎鍒槸-鈭歊^2-x^2,鈭歊^2-x^2,寰楀埌绉垎涓婁笅闄愩傜敱棰樼洰鍙煡锛歠(x,y)=1/蟺R^2 鑰岃竟缂樻鐜囧瘑搴Y(y)=鈭...
绛旓細x+y鈮1锛屽嵆鍗婂緞涓1鐨勫渾锛岄偅涔堟眰y鐨勮寖鍥达紝褰撶劧涔熷彲浠ョ浉绛夌殑锛屽嵆-鈭(1-x²)鈮鈮も垰(1-x²)銆傞殢鏈哄彉閲忔槸鍙栧兼湁澶氱鍙兘骞朵笖鍙栨瘡涓奸兘鏈変竴涓鐜囩殑鍙橀噺锛屽垎涓虹鏁ｅ瀷鍜岃繛缁瀷涓ょ锛岀鏁ｅ瀷闅忔満鍙橀噺鐨鍙栧间负鏈夐檺涓垨鑰呮棤闄愬彲鍒椾釜锛堟暣鏁伴泦鏄吀鍨嬬殑鏃犻檺鍙垪锛夛紝杩炵画鍨嬮殢鏈哄彉閲忕殑鍙栧间负鏃犻檺...
绛旓細锛圶,Y锛夌殑鑱斿悎姒傜巼瀵嗗害鏄痜(x,y)=1/蟺,x^2+y^2銆傛鐜囧瘑搴︾殑鐞嗚В锛氶鍏堬紝鎶奫F(x+螖x)-F(x)]锛徫攛鐨勫畾涔変负骞冲潎瀵嗗害锛岀劧鍚庡叾涓璅(x)灏辨槸鍒嗗竷鍑芥暟锛孾F(x+螖搴)-F(x)]锛徫攛閭ｄ箞灏辨槸骞冲潎鐨勬鐜囧瘑搴︿簡銆傜劧鍚庯紝鎴戜滑瀵逛笂寮忔潵鍙栨瀬闄愶紝杩欏氨鏄煇涓澶勭殑姒傜巼瀵嗗害浜嗭紝鍐嶇劧鍚巐im螖x瓒嬩簬0[F(...
绛旓細x+y鈮1锛屽嵆鍗婂緞涓1鐨勫渾锛岄偅涔堟眰y鐨勮寖鍥达紝褰撶劧涔熷彲浠ョ浉绛夌殑锛屽嵆-鈭(1-x²)鈮鈮も垰(1-x²)銆備緥濡傦細瑙ｏ細骞抽潰鍖哄煙D鏄竴涓钩琛屽洓杈瑰舰锛岄《鐐筪u鍒嗗埆涓哄師鐐(0,0)锛(1,0)锛(0,1)锛(-1,1)銆傛樉鐒跺叾闈㈢Н涓1脳1=1 鏁浜岀淮闅忔満鍙橀噺锛坸锛寉锛夌殑鑱斿悎姒傜巼瀵嗗害鍑芥暟涓 fX,Y(x,y)= {...
绛旓細杩欐槸涓潰绉负蟺R^2鐨鍦嗗舰锛屽潎甯冨湪鍦嗗唴(dx dy)鐨勬鐜囧间负1/蟺R^2銆 濡傛灉姹傝竟缂樺垎甯冪殑璇濓紝涔熷氨鏄眰f(x)鍜宖(y)锛岀敱瀵圭О鎬у彲鐪嬪嚭瀹冧咯褰㈠紡涓鏍 f(x) 鐨勫煎煙鏄-1鍒1, 鑰屽搴斾竴涓‘瀹歺鐨剏鐨勫煎煙鏄(-sqrt(1-x^2),sqrt(1-x^2)) 鎵浠(x) = 2sqrt(1-x^2)
绛旓細棣栧厛浣犺〃杩扮殑灏变笉瀵癸紝鍗曚綅鍦嗗唴鏈夊潎鍖鍒嗗竷锛屼笉瑕佺瀻鍐欍傚洜涓浜岀淮鍧囧寑鍒嗗竷鏄畾涔夊湪浜岀淮鍖哄煙鍐呯殑銆
绛旓細鍏蜂綋鍥炵瓟濡傚浘锛氬鏋浜岀淮闅忔満鍙橀噺X,Y鐨勫垎甯冨嚱鏁癋{x锛寉}涓宸茬煡锛岄偅涔堥殢鏈哄彉閲弜锛寉鐨勫垎甯冨嚱鏁癋𝗑{x}鍜孎ʏ{y}鍙敱F{x锛寉}姹傚緱銆傚垯F𝗑{x}鍜孎ʏ{y}涓哄垎甯冨嚱鏁癋{x锛寉}鐨勮竟缂樺垎甯冨嚱鏁般
