functions - Cannot evaluate differential in Mathematica

I have a small toy script in Mathematica that I am trying to use to evaluate the pdf of $Y$ where $Y=X^2$, and $X$ is uniformly distributed in $[0,a]$.

The script is,

assum = {a > 0}; 
Subscript[P, X][x_] := If[Inequality[0, Less, x, LessEqual, a], 1/a, 0];

Y[x_] := x^2

Subscript[F, Y][y_] := Integrate[If[Y[x] <= y, 1, 0]*Subscript[P, X][x], 
   {x, -Infinity, Infinity}]


Subscript[P, Y][y_] := D[Subscript[F, Y][y], y]

It shows the correct form of the pdf:

In[192]:=
Simplify[Subscript[P, Y][y], assum]

Out[192]=
Piecewise[{{1/(2*a*Sqrt[y]), y > 0 && a^2 >= y}}, 0]

However, it doesn't evaluate the expression correctly:

In[194]:=
Simplify[Subscript[P, Y][a*(a/2)], assum]

During evaluation of In[194]:= General::ivar:a^2/2 is not a valid variable. >>

Out[194]=
D[1/Sqrt[2], a^2/2]

The session, in pretty printing looks like this:

Answer

$P[y]$ is defined to be $\partial_yF[y]$. If you now call $P[2]$ for example, this is translated to $\partial_2F[2]$, which of course does not make any sense.

One way to get around this behavior is not taking the derivative with respect to y, but with respect to the first argument instead. The following example assigns f[y] to be the derivative of g[y]:

g[y_] := y^2
f[y_] := Derivative[1][g][y]
f[x]

2 x

Instead of Derivative[1][g][y] you could also have used the shorthand notation g'[y]. Read the explicit long version I used as an operator applied to multiple things: Derivative[1] takes a function and calculates its first derivative with respect to the first argument. Derivative[1][g] is that derivative (as a pure function), and Derivative[1][g][y] is that pure function applied to a value y.

The script above has one problem however: Every time you call f[x], it re-calculates the derivative, which can take some time if your function is more complicated and you need a lot of data points. If you use functions instead of patterns (i.e. no :=) you can also get around this problem:

g = #^2 &
f = g'
f[x]

#1^2 &
2 #1 &
2 x