(1)密度函數在積分區域的積分總是等於1，所以:
定積分(x從0到π) asin(x) dx = -acos(x) | x從0到π = 2a = 1，所以a = 1/2；
(2)直接使用定義，
EX =定積分(x從0到π)axsin(x) dx
=定積分(x從0到π)(-ax) dcos(x)
= -axcos(x) | x從0到π+定積分(x從0到π)acos(x) dx
= aπ+0 = aπ=π/2；
(3)
EX^2 =定積分(x從0到π) ax 2sin (x) dx
=定積分(x從0到π)(-ax ^ 2)dcos(x)
=(-ax ^ 2)cos(x)| x從0到π+定積分(x從0到π)axcos(x) dx
= aπ^2+definite積分(x從0到π)ax dsin(x)
= aπ^2-definite積分(x從0到π)asin(x) dx
= aπ2 = aπ^2-1 = π^2/2-1+0，所以
dx = π^2/2-1-π^2/4 = π^2/4-1；
(4)分布函數的定義:
F(x) =定積分(t從負無窮到x)f(x) dx
= 0，如果x & lt=0;
=定積分(t從0到x) asin (x) dx = (1-cos (x))/2，如果0
= 1，如果x & gtπ.