  Given a continuous, odd, reducible and semi-simple $2$-dimensional
representation $\bar\rho_0$ of $G_{\mathbb{Q},Np}$ over a finite field of odd
characteristic $p$, we study the relation between the universal deformation
ring of the pseudo-representation corresponding to $\bar\rho_0$
(pseudo-deformation ring) and the big $p$-adic Hecke algebra to prove that the
maximal reduced quotient of the pseudo-deformation ring is isomorphic to the
local component of the big $p$-adic Hecke algebra corresponding to $\bar\rho_0$
if a certain global Galois cohomology group has dimension $1$. This partially
extends the results of B\"{o}ckle to the case of residually reducible
representations. We give an application of our main theorem to the structure of
Hecke algebras modulo $p$. As another application of our methods and results,
we prove a result about non-optimal levels of newforms lifting $\bar\rho_0$ in
the spirit of Diamond-Taylor. This also gives a partial answer to a conjecture
of Billerey-Menares.
