Lighthill [1], in his Theory of Aerodynamic Noise, considered the noise from a pseudo-finite yet unbounded domain of compressible unsteady flow. The first application of this theory was given by Proudman [2] for the case of isotropic turbulence at low Mach numbers and high Reynolds numbers. More recently, Lilley [3] and Sarkar and Hussaini [4], using Direct Numerical Simulation (DNS), have reconsidered this problem, and evaluated for the first time the fourth order space-retarded time covariance which is central to Lighthill's theory for the determination of the acoustic radiated sound power. In this paper the previous work is extended to include the effects of a hot fluid in motion immersed in an external medium at rest. On the introduction of a simple hypothesis these results for the noise radiated from isotropic turbulence are used to predict the noise power radiated from a gaseous hot turbulent jet. The results are found to be qualitatively in agreement with far field experimental data on hot jets at subsonic and supersonic speeds, provided the jets are fully expanded and are devoid of shock waves. The theory has its origins in the 1950s following the publication of Lighthill's theory of aerodynamic noise, when Professor E. J. Richards, the author and their colleagues were striving to predict the noise from jet engines and establish methods for their noise reduction, without loss in performance.