Intutively and By Moment generating function technique, I can prove that independent bernoulli random variable's sum follows binomial distribution. But how can I prove that not by MGF technique?
Solved – How to prove that Bernoulli random variable’s sum is binomial distribution
bernoulli-distributionbinomial distributionmoment-generating-function
Best Answer
If you start from $$X_1,\ldots,X_n\stackrel{\text{i.i.d.}}{\sim}\mathcal{B}(p)$$ and define $$Y=X_1+\cdots+X_n$$ you can compute directly$$\mathbb{P}(Y=y)={n \choose y} p^y (1-p)^{n-y}\qquad y=0,1,\ldots,n$$ by a combinatoric argument.