Cumulative Distribution vs Probability Distribution

A cumulative distribution answers "at most" questions by stacking probabilities from the bottom up, so it climbs from 0 to 1 and never falls. If $X$ is the number of heads in two fair coin flips, with $P(0) = 0.25$, $P(1) = 0.50$, and $P(2) = 0.25$, the cumulative values are $0.25$, $0.75$, and $1$. In symbols, $F(x) = P(X \le x) = \sum_{x_i \le x} P(x_i)$ (the probability X is less than or equal to x). The cdf commands on a graphing calculator compute exactly this, which is why they answer "at most" questions but not "exactly" questions.

A probability distribution shows how the total probability of $1$ is shared among a random variable's possible values. For example, rolling a fair six-sided die gives each value from 1 to 6 a probability of $\frac{1}{6}$. For a discrete variable the probabilities must satisfy $0 \le p_i \le 1$ and $\sum p_i = 1$ (each probability is between 0 and 1, and they add to 1). Continuous variables use a density curve instead, where probability is the area under the curve.