两个均匀的 4 面骰子，其各个面上分别标有 '1'、'2'、'3' 和 '4'。掷出这两个骰子，记它们底面数字的差的绝对值为 $X$。

1. Tabulate the absolute differences for all possible outcomes of rolling two 4-sided dice.

Let the scores on the two dice be $D_1$ and $D_2$. The possible values for $X = |D_1 - D_2|$ are shown in the table:
$\begin{array}{|c|cccc|} \hline D_1 \setminus D_2 & 1 & 2 & 3 & 4 \\ \hline 1 & 0 & 1 & 2 & 3 \\ 2 & 1 & 0 & 1 & 2 \\ 3 & 2 & 1 & 0 & 1 \\ 4 & 3 & 2 & 1 & 0 \\ \hline \end{array}$
There are $4 \times 4 = 16$ equally likely outcomes.

2. Calculate the probability $\mathrm{P}(X=x)$ for each possible value of $X$.

Counting the frequencies of each possible absolute difference:

• $X = 0$ appears $4$ times: $\mathrm{P}(X=0) = \frac{4}{16} = \frac{1}{4}$
• $X = 1$ appears $6$ times: $\mathrm{P}(X=1) = \frac{6}{16} = \frac{3}{8}$
• $X = 2$ appears $4$ times: $\mathrm{P}(X=2) = \frac{4}{16} = \frac{1}{4}$
• $X = 3$ appears $2$ times: $\mathrm{P}(X=3) = \frac{2}{16} = \frac{1}{8}$

1. Recall or calculate the probability distribution of $X$ from part (i).

The possible outcomes for two 4-sided dice give a total of $4 \times 4 = 16$ equally likely pairs. The absolute differences $X$ and their probabilities are:

• $\mathrm{P}(X=0) = \frac{4}{16} = \frac{1}{4}$
• $\mathrm{P}(X=1) = \frac{6}{16} = \frac{3}{8}$
• $\mathrm{P}(X=2) = \frac{4}{16} = \frac{1}{4}$
• $\mathrm{P}(X=3) = \frac{2}{16} = \frac{1}{8}$

2. Calculate the expectation $\mathrm{E}(X)$.

3. Calculate the variance $\mathrm{Var}(X)$.

First, calculate $\mathrm{E}(X^2)$:

Then, use the variance formula: