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Write an expression for the average energy of a set of particles obeying Boltz-

mann statistics and each having energy E = bz2, where b is a constant and

z is a variable. Hence, show that the average energy per degree of freedom

for each particle is 1

2kBT; where kB is Boltzmann's constant. You should use

the standard integrals shown at the end of the question.

You are given:

[integral from -infinity to infinity]exp(-ax

^{2})dx = SQRT(PI/a)

and

[integral from -infinity to infinity]x

^{2}exp(-ax

^{2})dx = (1/2)SQRT(PI/a

^{3})

I've taken the average energy as [integral from -infinity to infinity]E.Aexp(-E/K

_{B}T)dE which gives:

[integral from -infinity to infinity]2b

^{2}z

^{3}exp(-bz

^{2}/K

_{B}T)dz by changing variable to z (dE/dz=2bz)

But this is not the standard result so I can't proceed!