In this work, we propose a method for solving a linear inverse problem where the operator depends on an unknown parameter. Our approach involves approximating the exact solution using a stochastic iterative Landweber-type scheme, while the optimal parameter value is determined by minimizing a least squares functional.  An exponential inequality will be established to demonstrate the almost complete convergence of the method, specifying both the rate of convergence and the confidence interval within which the solution resides.  Finally, we present a numerical application involving a Fredholm integral equation of the first kind, where the kernel depends on an unknown parameter.