  In this paper, we study the value-distributions of $L$-functions of
holomorphic primitive cusp forms in the level aspect. We associate such
automorphic $L$-functions with probabilistic models called the random Euler
products. First, we prove the existence of probability density functions
attached to the random Euler products. Then various mean values of automorphic
$L$-functions are expressed as integrals involving the density functions.
Moreover, we estimate the discrepancies between the distributions of values of
automorphic $L$-functions and those of the random Euler products.
