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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?

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    $\begingroup$ That's the definition of binomial distribution. In the Wikipedia article about binomial distribution (en.wikipedia.org/wiki/Binomial_distribution) you can see how its probability mass function and cumulative probability function are computed. $\endgroup$ – Pere Sep 18 '16 at 9:11
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    $\begingroup$ Is this question connected to the self-study tag in that it is connected with a book, exercise or homework assignment? In which case you would need to add the tag. $\endgroup$ – Xi'an Sep 18 '16 at 10:32
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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.

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