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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:

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  • $\begingroup$ 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. $\endgroup$ – 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 $x = 12$.

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).

  • $\begingroup$ 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? $\endgroup$ – whuber Jul 18 '17 at 16:59
  • $\begingroup$ @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. $\endgroup$ – Kodiologist Jul 18 '17 at 17:04
  • $\begingroup$ 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. $\endgroup$ – whuber Jul 18 '17 at 18:03
  • $\begingroup$ 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? $\endgroup$ – WorldGov Jul 19 '17 at 18:11
  • $\begingroup$ @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 $x > 12$. $\endgroup$ – Kodiologist Jul 19 '17 at 18:24

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