使用离散均匀分布作为模型
骰子一次投掷的点数 \(S\) 的概率分布如下：
The probability distribution for the score \(S\) on a single roll of a dice is:
| \(S\) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| \(P(S = s)\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) | \(\frac{1}{6}\) |
这是一个在集合 \(\{1,2,3,4,5,6\}\) 上的离散均匀分布。它是离散的，因为取值是离散的；它是均匀的，因为所有概率都是相同的。
This is an example of a discrete uniform distribution over the set of values (1,2,3,4,5,6). It is discrete because the values are discrete and it is uniform because all the probabilities are the same.
离散均匀分布的条件：
Conditions for discrete uniform distribution:
离散均匀分布的条件：
Conditions for Discrete Uniform Distribution:
在许多情况下，\(X\) 在集合 \(\{1,2,3,\ldots,n\}\) 上定义。在这种情况下，期望值和方差由以下公式给出：
In many cases, \(X\) is defined over the set \(\{1,2,3\ldots n\}\). In such cases, the mean and variance are given by the following:
离散均匀分布的期望值和方差
Mean and Variance of Discrete Uniform Distribution
\[\mathrm{E}(X) = \frac{n + 1}{2}\]
\[\operatorname{Var}(X) = \frac{(n + 1)(n - 1)}{12}\]
注意事项 / Watch Out：
记住，要使用这些公式，\(X\) 的取值必须是 \(1,2,3,\ldots,n\)。
Remember, in order to use these formulas, the values for \(X\) must be \(1,2,3,\ldots n\).
考试提示 / Examination Tip：
你不需要证明这些结果，但在考试中回答这类问题时可能会发现它们很有用。
You do not need to prove these results, but you may find them useful for the examination when answering questions of this type.
例 6.7.1 / Example 6.7.1：
从随机数字表中随机选择数字。
Digits are selected at random from a table of random numbers.
a) 求单个数字的均值和标准差。
a) Find the mean and standard deviation of a single digit.
b) 求特定数字落在均值一个标准差范围内的概率。
b) Find the probability that a particular digit lies within one standard deviation of the mean.
解答 / Solution：
在这个例子中，数字将是0到9（包括0和9）。
In this example the digits will be the numbers 0 to 9 inclusive.
令 \(R\) 表示这个随机变量，在集合 \(\{0,1,2,\ldots,9\}\) 上具有离散均匀分布。
Let \(R\) represent this random variable having a discrete uniform distribution over the set \(\{0,1,2,\ldots 9\}\).
令 \(X\) 表示一个随机变量，在集合 \(\{0,1,2,\ldots,10\}\) 上具有离散均匀分布。
Let \(X\) represent a random variable having a discrete uniform distribution over the set \(\{0,1,2,\ldots 10\}\).
\(X\) 和 \(R\) 之间有一个简单的关系，即 \(R = X - 1\)。
There is a simple relationship between \(X\) and \(R\), namely \(R = X - 1\).
通过引入随机变量 \(X\)，你可以使用 \(n = 10\) 的标准公式。
By introducing the random variable \(X\), you can use the standard formula for \(X\) with \(n = 10\).
a) \(\mathrm{E}(R) = \mathrm{E}(X - 1) = \mathrm{E}(X) - 1 = \frac{n + 1}{2} - 1 = \frac{10 + 1}{2} - 1 = \frac{11}{2} - 1 = 5.5 - 1 = 4.5\)
\(\operatorname{Var}(R) = \operatorname{Var}(X - 1) = \operatorname{Var}(X) = \frac{(n + 1)(n - 1)}{12} = \frac{(11)(9)}{12} = \frac{99}{12} = 8.25\)
因此：\(\sigma = \sqrt{8.25} = 2.87\)（取3位有效数字）。
Therefore: \(\sigma = \sqrt{8.25} = 2.87\) (3 s.f.).
b) 使用a中的 \(\sigma\) 值，要求的概率是：\(P(4.5 - 2.87 < R < 4.5 + 2.87) = P(1.63 < R < 7.37) = P(2 \leq R \leq 7) = \frac{6}{10} = \frac{3}{5}\)。
替代方法 / Alternative Method：
例16展示了如何使用序列 \(\{0,1,2,\ldots n\}\) 的公式以及 \(X\) 的函数公式来得到答案。你也可以通过写出概率分布并在没有任何先前计算的情况下求均值和方差来得到答案。这是一种可接受的替代方法，尽管是一种更长、更耗时的方法。
Example 16 shows how you can use the formulae for the series \(\{0,1,2,\ldots n\}\) and the formulae for the functions for \(X\) to get the answers. You could also get the answers by writing down the probability distribution and finding the mean and variance without any previous working out. This is an acceptable alternative for the examination, although a longer and more time-consuming method.
离散均匀分布的关键特征：
Key Features of Discrete Uniform Distribution:
学习建议 / Learning Tips：
熟练掌握离散均匀分布的标准公式对于快速计算某些概率问题很有帮助。多做练习有助于加深理解。
Mastering the standard formulas for discrete uniform distributions is helpful for quickly calculating certain probability problems. More practice helps deepen understanding.