Calculate the generating function gn of Xn and calculate g(t)=limn→∞gn(t) [closed]
Solution 1:
Assuming you're referring to the moment generating function, you get $$g_n(t)=E[e^{tX_n}]=E[e^{t(Y_1+Y_2+...+Y_n)}]=E[\prod_i^n e^{tY_i}] =\prod_i^nE[e^{tY_i}]$$ $$=(1-p_n +p_ne^t)^n$$ Where the $Y$'s are i.i.d Bernoulli$(p_n)$ random variables. With this we get $$\lim_{n\to \infty}g_n(t)=\lim_{n\to \infty}(1+\frac{\lambda e^t-\lambda}{n})^n=\exp(\lambda(e^t-1))$$