probability or statistics – Weird behavior of computing sum of two random variables using TransformedDistribution

I want to calculate the p.d.f of two discrete random variables. However, when I made a toy example that random variable was dist and I wanted to find p.d.f. of dist+dist, it failed.

dist = EmpiricalDistribution({2/9, 5/9, 2/9} -> {-1, 0, 1});
ya(t_) := PDF(TransformedDistribution(2*x, {x (Distributed) dist}), t);
ya /@ Range(-10, 10)
(*Output: {0, 0, 0, 0, 0, 0, 0, 0, 2/9, 0, 5/9, 0, 2/9, 0, 0, 0, 0, 0, 0, 0, 0} *)

The output seems wrong, because the p.d.f. of the sum of two random variables should be the convolution of these two. Hence the correct answer should be
enter image description here

Where did I make a mistake?