Let P(x) be the probability density function of a quantity x. Now consider a weighted averagel of P(x); i.e.,

Π(x) = ∫P(x+z)w(z)dz

where the weight function w(z) has the properties

∫w(z)dz = 1
and
∫zw(z)dz = 0

All integrals are between −∞ and +∞.

The expected value of x, E_Π{x}, according to the probabiliy density function Π is

E_Π{x} = ∫xΠ(x)dx = ∫∫xP(x+z)w(z)dzdx

(In the symbol E_Π{x} x stands for the name of the variable rather than a possible value.)

Now consider a change in the variable in the integral on the RHS of the above to x+z=y and hence dx=dy.

E_Π{x} = ∫∫(y−z)P(y)w(z)dzdy = ∫∫yP(y)w(z)dzdy − ∫∫zP(y)w(z)dzdy
= ∫[∫yP(y)dy]w(z)dz − ∫∫P(y)[zw(z)dz]dy
= ∫∫E_P{y}w(z)dz − ∫P(y)[∫zw(z)dz]dy

Because ∫w(z)dz = 1 and ∫zw(z)dz = 0 the above reduces to

E_Π{x} = E_P{y}