Question

设随机变量X的概率密度为f(x)=1/π(1+x^2),(－∞

Answer 1

当y≤0时,F(y)=P(Y

当y>0时, F(y)=P(Y

-√y)=P(X f(y)=F'(y)=[1/π(1+y)][1/(2√y)]-[1/π(1+y)][-1/(2√y)]=1/π(1+y)][1/√y]

2. E(arctanX)=∫(－∞,＋∞)arctanX[1/π(1+x^2)]dx=(1/2π)(arctanX)^2︱(－∞,＋∞)

=2(1/2π)(π/2)^2=π/4