  The effect of dimerization on the random antiferomagnetic Heisenberg chain
with spin 1/2 is studied by the density matrix renormalization group method.
The ground state energy, the energy gap distribution and the string order
parameter are calculated. Using the finite size scaling analysis, the
dimerization dependence of the these quantities are obtained. The ground state
energy gain due to dimerization behaves as $u^a$ with $a > 2$ where $u$ denotes
the degree of dimerization, suggesting the absence of spin-Peierls instability.
It is explicitly shown that the string long range order survives even in the
presence of randomness. The string order behaves as $u^{2\beta}$ with $\beta
\sim 0.37$ in agreement with the recent prediction of real space
renormalization group theory ($\beta =(3-\sqrt{5})/2 \simeq 0.382$). The
physical picture of this behavior in this model is also discussed.
