Theoretical analysis of the crest and extremal statistics of a square-law-derived random process is presented. The differences between the crest and the extremal statistics, particularly the bandwidth dependence of the latter, are illustrated for the Gaussian-derived case. The theory is applied to the statistical response of a uniform bar in tension. Particular attention is paid to the crest distributions which deviate from the integrated Rayleigh distributions in a predictable way at large amplitudes of vibration. Experimental distributions obtained from the strain response of a uniform bar forced in its fundamental mode by band-limited Gaussian white-noise signals are in excellent agreement with calculated values.