The CDF of the Binomial Distribution

Recall that if $X \sim B(n, p)$ is a binomial random variable counting the number of successes in $n$ independent trials with success probability $p$, then its probability mass function (PMF) is

$$P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}.$$

The cumulative distribution function (CDF) of $X$ gives the probability that $X$ is at most $k$:

$$F(k) = P(X \leq k) = \sum\limits_{i=0}^{k} \binom{n}{i} p^i (1-p)^{n-i}.$$

To evaluate the CDF at $k$, we sum the PMF values from $i = 0$ up to $i = k$.

For instance, let $X \sim B(3, 1/2)$. Then $P(X = 0) = \binom{3}{0}(1/2)^3 = 1/8$ and $P(X = 1) = \binom{3}{1}(1/2)^3 = 3/8$, so

$$F(1) = P(X \leq 1) = \dfrac{1}{8} + \dfrac{3}{8} = \dfrac{1}{2}.$$