Moment Generating Function

Question

show that g(t)=te^t cannot be a moment generating function.  

Answer 1

Let f(x)=xexp(x)

 

M(t) = ∫_–∞^{∞ exp(tx)f(x)dx}

 

= ∫ d[exp(x(t+1))/(t+1)]x         (do integration by parts)

 

M(t) = (x/(t+1))(exp(x(t+1)) - ∫dx(exp(x(t+1))/(1+t)   evaluated at x = -inf and +inf or

 

M(t) = (x/(t+1))(exp(x(t+1)) - exp(x(t+1))/(t+1)(t+1) evaluated at x = -inf and +inf.

 

M(t) = [(x/(t+1)) -1/(t+1)(t+1)]exp(x(t+1) evaluated at x = -inf and +inf.

 

This diverges.