The speed of a molecule in a uniform gas at equilibrium is a random variable V whose density function is given by S(1) = av²e-h*; v> 0, where b = m/(2kT') and k, T and m denote Boltzmann's constant, the absolute temperature, and the mass of the molecule, respectively. (a) Derive the distribution of W = mV² /2, the kinetic energy of the molecule. (b) Find E (W).

The speed of a molecule in a uniform gas at equilibrium is a random variable V whose density function is given by S(1) = av²e-h*; v> 0, where b = m/(2kT') and k, T and m denote Boltzmann's constant, the absolute temperature, and the mass of the molecule, respectively. (a) Derive the distribution of W = mV² /2, the kinetic energy of the molecule. (b) Find E (W).