Hello. Im trying to show if a distriubtion exists for a given moment generating function M(t)= t/(t-1) , ltl<1 . It says find one if it exists or prove there does not exist any. I dont know how to go about this question, but I feel its got a really easy solution. Thanks.
I know that just become the integral from 0 to inf ,of e^tx * the pdf, but I dont know why Im having such a hard time showing it is +inf for any t>0. Do I have to actually compute the integral? I was trying to find the limit as x->0 and also the limit when x->inf, and show it diverges at one end and converges at the other. But even then I am having trouble. Sorry for all the questions.
If X is a random variable with normal distribution then the random variable has a lognormal distribution.
I'll assume a standard normal distribution to make things simple. Then:
.
Since the exponent is a cubic for which the term is positive (if t > 0), the exponential must approach as .
Therefore .