We present an experimental study on the statistical properties of the injected power needed to maintain an inelastic ball bouncing constantly on a randomly accelerating piston in the presence of gravity. We compute the injected power at each collision of the ball with the moving piston by measuring the velocity of the piston and the force exerted on the piston by the ball. The probability density function of the injected power has its most probable value close to zero and displays two asymmetric exponential tails, depending on the restitution coefficient, the piston acceleration, and its frequency content. This distribution can be deduced from a simple model assuming quasi-Gaussian statistics for the force and velocity of the piston.