Question

设随机变量X~U(0,1),求Y=X^2的概率密度

网上看了很多类似的题 都看不懂= =求大神详解 什么求导看的晕乎乎

Answer 1

先求分布函数，对其求导，就获得概率密度函数；因为概率密度函数积分可以获得分布函数。

p(x) = 1, when 0<x<1.

F(0) = P(Y<=0) = P(X^2<=0) = 0, so p(y) = 0, when y<=0.

let y>=1, F(y) = P(Y<=y) = P(X^2<=y) = 1, so p(y) = F'(y) = 0, when y>=1;

let 0<y<1, F(y) = P(Y<=y) = P(X^2<=y) = P(0<X<sqrt(y)) = sqrt(y), so p(y) = 1/[2*sqrt(y)], when 0<y<1;

综上所述，

p(y) = 0, when y<=0;
p(y) = 1/[2*sqrt(y)], when 0<y<1;
p(y) = 0, when y>1.