This thesis presents significant advances in the understanding of the statistical properties of undulator radiation via two experiments carried out in the Integrable Optics Test Accelerator (IOTA) storage ring at Fermilab. The first experiment studied the turn-to-turn fluctuations in the power of the radiation generated by an electron bunch. The magnitude of these fluctuations depends on the 6D phase-space distribution of the electron bunch. The author presents the most complete theoretical description of this effect to date, and shows that it can be used to measure some electron bunch...