随机变量与离散分布Random Variables & Discrete Distributions
怎样把随机试验的结果变成可以画图和计算的数字？How can random outcomes become numbers that can be plotted and calculated?
前置知识：Prerequisites: B12：随机事件与概率。B12: random events and probability.
复习前置：Review prerequisites: B12 · 随机事件、概率与独立性Random Events, Probability & Independence
学完后你能：After this lesson you can: 把硬币序列映射为计数，计算并检查离散分布。Map coin sequences to counts, calculate a discrete distribution, and check it.
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连续抛三枚独立硬币，比较正面个数的理论分布和重复试验频率；调整单次正面概率。Toss three independent coins and compare the distribution of head counts with observed frequencies; adjust the probability of heads.
从观察到解释From observation to explanation
从结果到数字From outcomes to numbers
连续三次抛硬币的原始结果记录顺序，例如正反正。定义随机变量 X 为正面个数，则正反正和正正反都对应 X=2。A raw result of three tosses records order, such as heads–tails–heads. Define X as the number of heads: heads–tails–heads and heads–heads–tails both map to X=2.
把相同数字合并Combine equal values
三枚公平且独立的硬币有 8 个等可能顺序。X=0、1、2、3 分别包含 1、3、3、1 个顺序，因此概率是 1/8、3/8、3/8、1/8。Three independent fair coins have 8 equally likely sequences. X=0, 1, 2, 3 contain 1, 3, 3, 1 sequences, so their probabilities are 1/8, 3/8, 3/8, 1/8.
分布与观察Distribution and observation
分布是所有可能数值及对应概率；一次观察只是其中一个值。一次成功或失败称为伯努利变量，独立同概率的多次成功计数服从二项分布。A distribution lists possible values and their probabilities; one observation is only one value. One success/failure is a Bernoulli variable; the success count in repeated independent identical trials has a binomial distribution.
公式与符号Formula and symbols
X 把每个原始结果映射成一个数；k 是可能取值，pₖ 是取到 k 的概率；对全部取值求和必须为 1。X maps each raw outcome to a number; k is a possible value and pₖ its probability. Probabilities summed over all values must equal 1.
留意这个区别Watch for this
正面个数丢失了顺序信息。所有概率必须非负且总和为 1；频率在有限样本下不必等于理论概率。A head count loses order information. Probabilities must be nonnegative and sum to 1; finite-sample frequencies need not equal theoretical probabilities.
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