设随机变量X服从参数λ的指数分布，令Y=[X]+1，求Y的概率函数 设随机变量X服从参数为λ的指数分布，求Y=X2的概率密度函数...
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\u6240\u4ee5Fy(y)=P(Y=e^x<y)=P(0<=x<=lny)
\u6240\u4ee5Fy(y)\u662f\u4e0a\u5f0f\u7684\u79ef\u5206\uff0c\u4e3a1-1/y\uff0c(y>=1)
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\u56e0\u4e3a
P_Y(y)=F(Yo}
\u4ece\u800c
f_Y(y)={0,yo}
x>0时, P{X<x}=1-e^(-λx)
F(y)=P{Y<y}=P{X+1<y}=P{X<y-1}
y<=1时, F(y) = P{X<y-1} = 0, f(y) = F'(y) = 0.
y>1时, F(y) = P{X<y-1} = 1-e^[-λ(y-1)], f(y) = F'(y) = -e^[-λ(y-1)]*(-λ) = λe^[-λ(y-1)]
Y=X+1 的概率密度函数为,
y<=1时, f(y)=0,
y>1时, f(y)=λe^[-λ(y-1)]
Y=X+1的概率分布函数为,
y<=1时, F(y)=P{Y<y} = 0,
y>1时, F(y)=P{Y<y} = 1 - e^[-λ(y-1)]
绛旓細銆愮瓟妗堛戯細鐢闅忔満鍙橀噺X鏈嶄粠鍙傛暟涓位鐨勬寚鏁鍒嗗竷锛屽緱锛屼簬鏄紝鑰岋紝瑙ｅ緱位=.
绛旓細P(X>1)=e^(-位)=e^(-2),鍒櫸=2
绛旓細Y=X+1 鐨勬鐜囧瘑搴﹀嚱鏁颁负,y<=1鏃, f(y)=0,y>1鏃, f(y)=位e^[-位(y-1)]Y=X+1鐨勬鐜囧垎甯冨嚱鏁颁负,y<=1鏃, F(y)=P{Y<y} = 0,y>1鏃, F(y)=P{Y<y} = 1 - e^[-位(y-1)]
绛旓細P{x>鈭欴{x)}=1-P{x<=鈭欴{x)}=1-int{(0-->1/位)位e^(-位x)} =1-{-e^(-位x)}(0-->1/位)=1-(1-1/e)=1/e
绛旓細鏍规嵁鎸囨暟鍒嗗竷鐨勬鐜囧瘑搴﹀嚱鏁帮紝璁鹃殢鏈哄彉閲 X 鏈嶄粠鍙傛暟涓 位 鐨勬寚鏁板垎甯冿紝鍒欏叾姒傜巼瀵嗗害鍑芥暟涓猴細f(x) = 位e^(-位x)锛屽綋 x 鈮 0 鏃 鍥犳锛屽綋 X 鏈嶄粠鍙傛暟涓 0.1 鐨勬寚鏁板垎甯 e(0.1) 鏃讹紝鏈夛細f(x) = 0.1e^(-0.1x)锛屽綋 x 鈮 0 鏃 瑕佽绠 P{5 < X < 30}锛屽嵆璁＄畻 X 鍦ㄥ尯闂...
绛旓細浣犲彲鑳藉叕寮忚閿欎簡銆鎸囨暟鍒嗗竷瀹為檯涓婄殑F(x)鏄1-e^(-x/胃)锛岃屼笉鏄1-e^(-x/位)銆傚湪鎸囨暟鍒嗗竷涓紝胃鐨勫畾涔夊疄闄呬笂灏辨槸1/位銆傚洜姝わ紝绛旀瑕佸掕繃鏉ワ紝灏辨槸2銆
绛旓細鈭闅忔満鍙橀噺X鏈嶄粠鍙傛暟涓位鐨勬寚鏁鍒嗗竷鈭碭鐨勬鐜囧瘑搴︿负锛歠(x)锛澪籩?位x锛寈锛00锛寈鈮0涓擠X锛1位2鈭碢{X锛濪X}=P{X锛1位}=1-P{X鈮1位}=1?鈭1位?鈭瀎(x)dx=1?鈭1位0位e?位xdx=1+e?位x|1位0锛1e
绛旓細鎵浠y(y)鏄笂寮忕殑绉垎锛屼负1-1/y锛(y>=1)鎵浠y(y)鏄笂寮忕殑瀵兼暟锛屼负1/y^2锛(y>=1)锛屽叾浣欎负0銆傜敱浜闅忔満鍙橀噺X鐨勫彇鍊 鍙彇鍐充簬姒傜巼瀵嗗害鍑芥暟鐨勭Н鍒嗭紝鎵浠ユ鐜囧瘑搴﹀嚱鏁板湪涓埆鐐逛笂鐨勫彇鍊煎苟涓嶄細褰卞搷闅忔満鍙橀噺鐨勮〃鐜般傝繛缁瀷鐨勯殢鏈哄彉閲忓彇鍊煎湪浠绘剰涓鐐圭殑姒傜巼閮芥槸0銆備綔涓烘帹璁猴紝杩炵画鍨嬮殢鏈哄彉閲忓湪鍖洪棿涓...
绛旓細X鏈嶄粠鍙傛暟位 涓鐨勬寚鏁鍒嗗竷锛屽垯: EX=1/位,X鏈夊垎甯冨嚱鏁帮細F(x)=1-e^(-位 x) , x>=0 ;浜庢槸 P(X>EX)= 1- P(X<=EX)= 1- P(X<=1/位)=1-F(1/位)=1-[1-e^(-位*1/位)]= e^(-1)
绛旓細鍥犱负 P_Y(y)=F(Y<=y)=F(X^2<=y)={0,y<=0 ; F_X(y^1/2)-F_X(-y^1/2),y>o} 浠庤 f_Y(y)={0,y<=0 ; f _X(y^1/2)*1/2*y^(-1/2)=1/2*位*e^(-位*y^1/2)*y^(-1/2),y>o}
