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 $0<\lambda<n$. 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.
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$\begingroup$ 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. $\endgroup$– User1865345Commented Mar 11, 2023 at 21:03
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$\begingroup$ @User1865345 thanks for your comment. I modified the symbolism but I do not have any idea about how to start. $\endgroup$– SinaCommented Mar 11, 2023 at 21:12
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1 Answer
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What should be the approach here?
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 $1/n.$
We can hope the limiting calculation could be implemented using elementary calculus techniques.
Can we now formally start to solve the problem? To see what happens? What could go wrong?
answered Mar 11, 2023 at 21:30
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$\begingroup$ how t to determine the asymptotic distribution of $Y_n$ $\endgroup$– SinaCommented Mar 12, 2023 at 5:41
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$\begingroup$ 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. $\endgroup$ Commented Mar 12, 2023 at 21:38