设随机变量x的概率密度为f(x)=e^-x,x>0,f(x)=0,其它,求y=x^2的概率密度
F(y)=P(Y绛旓細鐢X鐨勬鐜囧瘑搴鍙緱锛孹鐨勫垎甯冨嚱鏁颁负锛歺鈮0鏃讹紝F锛坸锛=P{X鈮}= 0锛0锛渪锛1鏃讹紝F锛坸锛=P{X鈮}=x^2锛泋鈮1鏃讹紝F锛坸锛=P{X鈮}=1銆傛墍浠ワ紝鍦ㄤ竴娆¤瀵熶腑浜嬩欢{X鈮1/2}鍑虹幇鐨勬鐜囦负锛1/4銆
绛旓細锛2锛夌敱F(x)锛濃埆x−鈭f(x)dx锛屽緱褰搙鈮0鏃讹紝F锛坸锛=0锛涘綋0锛渪锛1鏃讹紝F(x)锛濃埆x04x3dx锛漻4锛涘綋x鈮1鏃讹紝F锛坸锛=1鈭碏(x)锛1锛寈锛1x4锛0鈮鈮10锛寈锛0 锛3锛夌敱浜嶱锛X锛瀉锛=1-P锛圶鈮锛=1-P锛圶锛渁锛=P锛圶锛渁锛夆埓P锛圶锛渁锛=0.5鈭粹埆a04x3dx锛漚4锛0.5鈭...
绛旓細x=(+or-y^0.5)锛寍Jacobian|=|dx/dy|=1/2y^-0.5 f(y)=(0.5y^-0.5) (fx(y^0.5)+fx(-y^0.5))= (0.5y^-0.5)(e^(y^0.5)+e^(-y^0.5))浠绘剰鐨闅忔満鍙橀噺x锛寉=x^2鐨勫垎甯冮兘鏄(0.5y^-0.5)(fx(y^0.5)+fx(-y^0.5))涓嬫鐩存帴濂楄繖涓叕寮忓氨濂斤紝涓婇潰鐨勮瘉鏄庡浜...
绛旓細瀵姒傜巼瀵嗗害绉垎锛岀粨鏋滀负鈭f(x)dx=[ax+(bx^3)/3], 鍦ㄩ浂鍒颁竴鍖洪棿鍐咃紝寰楀埌a+b/3=1锛涘钩鍧囧 鈭玣(x)*xdx=(ax^2)/2+(bx^4)/4, 鍦ㄩ浂鍒颁竴鍖洪棿鍐咃紝寰楀埌a/2+b/4=3/5锛涙晠a=0.6, b=1.2. 鏂瑰樊 鈭(f(x)-3/5)^2dx, 浠ｅ叆a,b, 鍦ㄩ浂鍒颁竴鍖洪棿鍐咃紝寰楀埌0.288....
绛旓細鈭玔-鈭,鈭瀅 f(x)dx=ax^2/2|(0,2)=2a=1 a=1/2 锛1锛夊垎甯冨嚱鏁F(x)=0, x<0 F(x)=x^2/4,0<=x<=2 F(x)=1, x>2 锛2锛塅y(y)=0,y<0 Fy(y)=P(Y<=y)=P(X<=y/2)=y^2/16, 0<=y<=4 Fy(y)=1,y>4 鍥犳fy(y)=y/4, 0<=y<=4 鍏朵粬涓0.濡傛湁鎰忚...
绛旓細瑙ｏ細E(x)=鈭(-鈭,+鈭)f(x)xdx=a/3+b/2=7/12 鈭(-鈭,+鈭)f(x)dx=a/2+b=1 a=1,b=1/2 E(x^2)=鈭(-鈭,+鈭)f(x)x^2dx=a/4+b/3=5/12 D(x)=E(x^2)-E^2(x)=11/144 濡傛湁鎰忚锛屾杩庤璁猴紝鍏卞悓瀛︿範锛涘鏈夊府鍔╋紝璇烽変负婊℃剰鍥炵瓟锛
绛旓細A=2銆璁鹃殢鏈哄彉閲廥鍏锋湁姒傜巼瀵嗗害fX(x)锛-鈭<x<鈭烇紝鐢辫鍑芥暟g(x)澶勫鍙涓旀亽鏈塯'(x)>0锛堟垨鎭掓湁g'(x)<0锛夛紝鍒橸=g(X)鏄繛缁瀷闅忔満鍙橀噺銆傜Н鍑烘潵鐨勫嚱鏁版槸ax^3/4,绉垎鍊兼槸a*1^3/4-a*0^3/4=a/4銆傚f(x)=Ax鍦0鍒1涓婄Н鍒嗭紝 寰楀埌0.5A=1锛岃В寰桝=2銆
绛旓細浣滅Н鍒 鈭(0,1) 位 dx+鈭(1,3) 位(x-1)dx=1. 瑙ｅ緱位=1/3. x/3锛 0鈮鈮1 (2)瀵f(x)绉垎鍗冲緱F(x)=x²/6-x/3锛1鈮鈮3 c锛 鍏跺畠.
绛旓細X鐨勫瘑搴鍑芥暟鏄fX(x)=1/鈭2蟺*e^(-x^2/2)閭ｄ箞 FY(y)=P(Y<=y)=P(e^(-x)<=y)=P(x>=-lny)=1-P(x< -lny)=1-FX(-lny) FX(x) FY(y)琛ㄧずXY鐨勫垎甯冨嚱鏁 鎵浠鐨勫瘑搴﹀嚱鏁版槸锛歠Y(y)=FY'(y)=(1-FX(-lny))'=(-1)*(FX(-lny)'*(-lny)'=(-1)*fX(-lny)*(-1/...
绛旓細2012-09-13 璁鹃殢鏈哄彉閲廥鐨勬鐜囧瘑搴︿负f(x)={x,0<=x<1;2-x... 22 2012-04-22 璁鹃殢鏈哄彉閲廥鐨勬鐜囧瘑搴︿负f(x)={鈶燼x+1,0鈮鈮2 ... 8 2013-01-22 璁鹃殢鏈哄彉閲廥鐨勬鐜囧瘑搴﹀嚱鏁颁负f(x)={ax² 0... 7 2014-03-18 宸茬粡闅忔満鍙橀噺X鐨勬鐜囧瘑搴︿负f(x)=Ax,0<x<1;0,鍏... 2 2018-...
