| discrete distribution |
Yes
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The roll of a fair die has a discrete distribution, because the roll takes one of the $6$ values $1, 2, \dots, 6$.
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| continuous distribution |
No
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The roll of a fair die does not have a continuous distribution, because each of the values $1, \dots, 6$ has probability $\frac{1}{6}$.
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| mixed distribution |
No
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The roll of a fair die does not have a mixed distribution, because all of its probability is on single values.
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| singular distribution |
No
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The roll of a fair die does not have a singular distribution, because single values have positive probability.
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| probability mass function |
$p(k) = \frac{1}{6}$ for $k = 1, \dots, 6$
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The roll of a fair die has the probability mass function $p(k) = \frac{1}{6}$, because each of the $6$ outcomes is equally likely.
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| cumulative distribution function |
$F(x) = \frac{\lfloor x \rfloor}{6}$ for $0 \leq x \leq 6$
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The cumulative distribution function of a fair die roll is a staircase, rising by $\frac{1}{6}$ at each of $1, 2, \dots, 6$, because $P(X \leq x)$ counts the outcomes at most $x$.
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| expected value |
$\mathbb{E}[X] = 3.5$
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$\mathbb{E}[X] = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5$. Note: The expected value is not a possible outcome.
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| median |
$\frac{3 + 4}{2} = 3.5$
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The expected value and the median of a fair die roll are both $3.5$, which is not a possible outcome.
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| mode |
Every outcome, $1$ through $6$
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Every outcome is equally likely.
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| variance |
$\operatorname{Var}(X) = \mathbb{E}[X^2] - 3.5^2 = \frac{91}{6} - \frac{49}{4} = \frac{35}{12} \approx 2.92$
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| standard deviation |
$\sqrt{35 / 12} \approx 1.71$
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