# 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

## 1 Answer

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 $x = 12$.

The table is miscaptioned. It shows the cumulative distribution function, like the figure, not the probability density function (also known, for discrete variables, as the probability mass function).

• The table's title is incorrect in every important respect: two dice have no probability density function and the values in the table sketch out parts of its CDF. Would noticing these errors change parts of your answer? – whuber Jul 18 '17 at 16:59
• @whuber Good catch, the table is for the CDF, not the PDF. I disagree that dice have no PDF. A PMF is just a PDF where the measure that the integral is taken with respect to (for computing the corresponding CDF) is the counting measure. – Kodiologist Jul 18 '17 at 17:04
• I won't quibble over PDF vs PMF, because it's a matter of definition. I think like you about this, but most of the definitions I have encountered insist that a PDF be absolutely continuous with respect to Lebesgue measure. – whuber Jul 18 '17 at 18:03
• 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