Question

设二维随机变量x y在由y=1-x^2 与y=0所围区域d上服从均匀分布 写出x y的概率密度与边缘密度概率

Answer 1

面积=∫(-1~1)(1-x²) dx
=x-x³/3|(-1~1)
=2-2/3
=4/3

故概率密度f(x,y)=3/4 (0<=y<=1-x²)
=0 (else)
fx(x)=∫(0~1-x²)3/4dy
=3(1-x²)/4 (-1<=x<=1)
=0(else)

fy(y)=∫(-根号(1-y)~根号(1-y)) 3/4 dx
=6根号(1-y)/4 (0<=y<=1)
=0(else)