years, $P(\Gamma)$ has been a subject of investigations [@disordered; @GG:00] for a, simple but much studied, discrete tight-binding one dimensional random chain which is coupled to a perfect lead at one side. A numerical study [@GG:00] shows that in a broad range of $\Gamma$, $P(\Gamma)\sim \Gamma^{-\gamma}$, where the exponent $\gamma$ is very close to $1$. Intuitively the $1/\Gamma$ behaviour can be deduced by assuming a uniform distribution for the localization centers of exponentially localized states [@Casati:99]. However, from analytic point of view one usually considers an infinitely long chain in which case the average density of resonances (DOR) has a well defined limit. For a finite size system, the difference between the DOR and $P(\Gamma)$ is the normalization by the system size [@KS:06; @KS:08]. Recently, Kunz and Shapiro have derived analytic expression of the DOR for a semi-infinite disordered chain [@KS:08]. They have obtained an exact integral representation of the DOR which is valid for arbitrary lead-chain coupling strength. This has been further simplified for small lead-chain coupling strength where a universal scaling formula is found. In this limit they have proved the $1/\Gamma$-behavior of the DOR [@KS:06; @KS:08]. Besides, for the continuous limit of this model an integral representation 