# How do I interpret discontinuous graphs of a probability distribution function (discrete)? Why not simply display it as points?

The graph shows the cumulative probability distribution function for the sum of two fair dice. Why is the probability (y axis) of a given random variable (x axis)shown as lines, and not simply as points?

Also, why is there a large line that extends at x = 12? You can find the table here:

• cdfs are defined over the whole line. You can readily calculate $P(X\leq 6.75)$ for the sum of two fair dice, for example. – Glen_b Jul 18 '17 at 23:19

Because it's a cumulative distribution function. Take, for example, the line from $x = 2$ to $x = 3$. If you just drew a point at $(2, \tfrac{1}{36})$ instead of drawing a line segment, that would mean that the function isn't defined between $2$ and $3$, when in fact it is. For example, the value of the function at $2\tfrac{1}{2}$ is $\tfrac{1}{36}$. Likewise, the value of the function is $1$ for all $x ≥ 12$, so the graph has a ray starting at $x = 12$.
• Thank you, that helped. You say that the value of the function is 1 for all $x ≥ 12$, but you actually meant that the value of the function is 1 for all $x ≤ 12$, right? – WorldGov Jul 19 '17 at 18:11
• @WorldGov No. The CDF $F(x)$ is defined to give you the probability of drawing a value less than or equal to $x$. Since the most a pair of dice can roll is $12$, $F(12) = 1$, and also $F(x) = 1$ for all $x > 12$. – Kodiologist Jul 19 '17 at 18:24