We present an algorithm to generate wideband speech from a narrowband version of the same. The main body of the algorithm is a Statistical Recovery Function (SRF), which predicts the highband spectrum based on the narrowband spectrum. Assuming that bandpass portions of the speech are generated completely by a fixed number of random sources, the SRF explores the dependency among the random sources. The performance of the algorithm has been measured both in terms of spectral distortion and spectral signal-to-noise ratio (SNR). We obtained a 3 dB gain in SNR for the reconstructed wideband speech as compared to the narrowband speech. Informal perceptual experiments indicate a significant preference for the reconstructed speech.