已知二维随机变量(X,Y)的概率密度为 f(x,y)={ x^2+1/3xy 0
fx(x)=∫(0->2) x^2+1/3xy dy=2x^2+2/3x 0绛旓細1銆佹眰闅忔満鍙橀噺X鐨勫瘑搴X(x)锛岃竟娌垮垎甯 fX(x)={e^(-y)锛0<x<y锛泏0 2銆佹鐜囧瘑搴﹀嚱鏁癴(x,y)鍦ㄧ洿绾縳=0,y=x,y=-x+1鎵鍥寸殑涓夎褰㈠尯鍩熺殑浜岄噸搴︾Н鍒嗭紝缁撴灉鏄1+e^(-1)-2e^(-1/2)3銆佹潯浠跺垎甯冿紝搴旇鍐欐垚 fX(x|Y=y)鑰岄潪f尉(x|畏=y)锛岃〃绀篩=y鐨勬潯浠跺垎甯冿紝鎸夐鐩剰鎬濓紝姝ゅy...
绛旓細瑙佷笅鍥撅細
绛旓細Cov(X,Y)=E(XY)-E(X)E(Y)=1/6-(5/12)²=-1/144銆傚洜涓哄垎甯冨嚱鏁 F(x0,y0)=P{X<x0&&Y<y0} 涓嶇x0锛寉0璋佸ぇ璋佸皬锛屾寚鐨勬槸 Y=y0鐩寸嚎浠ヤ笅銆乆=x0鐩寸嚎涔嬪彸鍖哄煙鍐呯殑绉垎锛岃岃繖涓尯鍩熷唴铏界劧 x>y澶勫瘑搴﹀嚱鏁颁负0锛屼絾杩樻槸鏈 x<y鐨勭偣鐨勩備緥濡傦細璁浜岀淮闅忔満鍙橀噺(X,Y)鐨勬鐜囧瘑搴...
绛旓細瀵逛簬浜岀淮杩炵画鍙橀噺鐨勫垎甯冨嚱鏁癋(x锛寉)锛屼竴鑸簲鐢ㄥ叾姒傜巼瀵嗗害鍑芥暟f(x锛寉)鐨瀹氱Н鍒嗘眰瑙ｏ紱瀵逛簬闈炶繛缁彉閲忥紝闇瑕佸垎鍒疮鍔犳眰寰椼愪笌涓缁闅忔満鍙橀噺鐨勬眰娉曠浉浠裤戙傗埓鏈涓紝褰搙鈭(0,鈭)銆亂鈭(0锛屸垶)鏃讹紝鍒嗗竷鍑芥暟F(x锛寉)=鈭(-鈭烇紝x)du鈭(-鈭,y)f(u锛寁)dv=鈭(0锛寈)du鈭(-0锛寉)2e^(-2u-v...
绛旓細瀵筬(x,y)鍦ㄥ尯鍩焫+y
绛旓細璇︾粏姝ラ鏄細鈶犲厛姹傚嚭X銆乊鐨勮竟缂樺垎甯冦傛寜鐓у畾涔夛紝fX(x)=鈭(-鈭,鈭)f(x,y)dy銆傗埓fX(x)=鈭(0,x)15xy²dy=5x^4锛屽叾涓0<x<1銆傚悓鐞嗭紝fY(y)=鈭(y,1)15xy²dx=15(y²-y^4)/2锛屽叾涓0<y<1銆傗憽姹傚嚭E(X)銆丒(Y)銆丒(X²)銆丒(Y²)銆侲(X)=鈭(...
绛旓細褰0<y<x鏃讹紝F(x,y)=P(X<=x,Y<=y)=鈭埆xe^(-y锛塪xdy=鈭(0,y) xdx鈭(x,y) e^(-y锛塪y=1-(y+1)e^(-y)-y^2/2*e^(-y锛夊綋x锛寉鍙栧叾瀹冨兼椂锛孎(x,y)=0 鍒嗗竷鍑芥暟鏄闅忔満鍙橀噺鏈閲嶈鐨勬鐜囩壒寰侊紝鍒嗗竷鍑芥暟鍙互瀹屾暣鍦版弿杩伴殢鏈哄彉閲忕殑缁熻瑙勫緥锛屽苟涓斿喅瀹氶殢鏈哄彉閲忕殑涓鍒囧叾浠栨鐜囩壒寰...
绛旓細瀵筬(x,y)鍦ㄥ尯鍩焫+y<=1绉垎鍗冲彲锛屾湁鏁堝尯鍩熸槸涓瓑鑵扮洿瑙掍笁瑙掑舰涓夐《鐐(0 0)(1 0)(1/2 1/2)鎸墄=1/2鍒嗕袱鍧椼傜劧鍚庡寲绱绉垎鍢涳紝鍏堝y绉垎 宸﹁竟鍗婂潡锛寉绉垎绾 0鍒皒锛寈绉垎绾0鍒1/2 鍙宠竟鍗婂潡锛寉绉垎绾0鍒1-x,x绉垎绾1/2鍒1 ...
绛旓細EX=鈭埆[0<=y<=x<=1] xf(x,y)dxdy=鈭玔0->1]鈭玔0->x] 12xy²dydx=4/5 EY=鈭埆[0<=y<=x<=1] yf(x,y)dxdy=鈭玔0->1]鈭玔0->x] 12y³dydx=3/5 E(X²+Y²)=鈭埆[0<=y<=x<=1] (x²+y²)f(x,y)dxdy=鈭玔0->1]鈭玔0->x...
绛旓細棰樹笉闅撅紝璁＄畻澶鏉傦紝缁欎綘涓濊矾 (1) 1=鈭(0,+鈭)dx鈭(x,+鈭)ke^(-y)dy=k (2)P(X+Y<=1)=鈭(0,1/2)dx鈭(x,1-x)e^(-y)dy (3)fX(x)=鈭(x,+鈭)e^(-y)dy=e^(-x) (x>0)(4)fY(y)=鈭(0,y)e^(-y)dx=ye^(-y) (y>0)f(x|y)=f(x,y)/fY(...
