How would I find the cumulative distribution function of a binomial? I know I'd have to integrate it with its given parameters but how would someone go about doing that?
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$\begingroup$Binomial distribution is discrete, so you can't integrate it, but rather sum. This is what you should look into. If $X \sim Binomial(n,p)$, then CDF of $X$ is $$ P(X\leq m)=\sum_{k=0}^{m}\binom{n}{k}p^{k}(1-p)^{n-k} $$ This expression does not exist in closed form, since partial sum of rows of Pascal triangle do not exist in closed form. Depending on what exactly you need you can find various approximations to this sum, e.g. using Central Limit Theorem for large $n$, since Binomial RV is a sum of Bernoulli RVs.
EDIT: OK, as the OP specified the parameters are $n=3,p=\frac{1}{2}$, so the CDF is just $$ P(X \leq m)=\sum_{k=0}^{m}\binom{3}{k}\frac{1}{2^{k}}\bigg(1-\frac{1}{2}\bigg)^{3-k}=\frac{1}{8}\sum_{k=0}^{m}\binom{3}{k} $$
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