Question

We have n = 100 many random variables xi 's, where the xi 's are independent and identically distributed bernoulli random variables with p = 0.5 (e(xi)=p and var(xi)=p(1-p)).
a. what distribution does pn i=1 xi follow exactly (sum bernoulli random varaibles)? state the type of distribution and what the parameter is

Answer 1

Recall that for a random variable $X$ following a Bernoulli distribution $\mathrm{Ber}(p)$, we have the moment-generating function (MGF)

$M_X(t)=(1-p+pe^t)$

and also recall that the MGF of a sum of i.i.d. random variables is the product of the MGFs of each distribution:

$M_{X_1+\cdots+X_n}(t)=M_{X_1}(t)\times\cdots\times M_{X_n}(t)$

So for a sum of Bernoulli-distributed i.i.d. random variables $X_i$, we have

$M_{\sum\limits_{i=1}^nX_i}(t)=\displaystyle\prod_{i=1}^n(1-p+pe^t)=(1-p+pe^t)^n$

which is the MGF of the binomial distribution $\mathcal B(n,p)$. (Indeed, the Bernoulli distribution is identical to the binomial distribution when $n=1$.)