This work explores the dynamics of a three-dimensional four-bar mechanical linkage subject to random external forcing. The Lagrangian formulation of the equations of motion are index-3 stochastic differential-algebraic equations (SDAE) that describe the time evolution of the sample paths of the generalized coordinates, velocities, and Lagrange multipliers as stochastic processes. We numerically solve the SDAEs using two different approaches: inverse dynamics, Case Study 1, via independent, successive solution of the nonlinear equations for each kinematic variable, where the time evolution of one generalized coordinate is prescribed; and direct dynamics, Case Study 2, via direct solution of the SDAEs in the index-1 formulation, using fourth-order stochastic backward differentiation formula (BDF) with modified Newton iteration and position and velocity stabilization, where the (deterministic) input driving torque is prescribed.