# Limiting value of the moment generating function

Suppose that the discrete random variable $$X_{n}$$has a geometric distribution given by $$f_{X_n}(x_n)=P_n{(1-P_n)}^{x_n}$$ where $$x_n={0,1,2,3,}$$ and $$P_n\ =\ \frac{\lambda}{n}$$ for $$0<\lambda. Find the limiting value of the moment-generating function of $$Y_n= \frac{X_n}{n}$$ as $$n\rightarrow\infty$$ and use this result to determine the asymptotic distribution of $$Y_n$$. I found this question in a book titled Exercises and Solutions in Statistical Theory (Exercise 3.23) and really appreciate it if you can help me.

• I am unaware of such symbolism $f_{X_n}(X_n).$ It should be $f_{X_n}(x_n).$ Mainly you should show some attempts to solve the problem in hand and tell us where you stumbled in attempting the same. Mar 11 at 21:03
• @User1865345 thanks for your comment. I modified the symbolism but I do not have any idea about how to start.
– Sina
Mar 11 at 21:12

We know if $$X\sim \textrm{Geom}(p), ~M_X(t) = p(1-qe^t)^{-1}.$$ Also we know for a constant $$c,~M_{cX}(t) = M_X(ct).$$ Here $$c$$ should be $$1/n.$$
• how t to determine the asymptotic distribution of $Y_n$
• One approach would be to identify the distribution by inspection from $M_{Y_n}(t).$ The most general approach would be to use the inversion formula. Mar 12 at 21:38