Discrete Random Variables | 离散型随机变量

📚 Discrete Random Variables | 离散型随机变量

In probability theory, a random variable is a function that assigns a numerical value to each outcome in a sample space. A discrete random variable takes on a finite or countably infinite number of possible values. Understanding discrete random variables is essential for solving problems in games of chance, statistics, and everyday decision-making.

在概率论中，随机变量是一个将样本空间中的每个结果赋予一个数值的函数。离散型随机变量只能取有限个或可数无限个值。理解离散型随机变量对于解决机会游戏、统计和日常决策问题至关重要。

1. What is a Discrete Random Variable? | 什么是离散型随机变量？

A discrete random variable, often denoted by a capital letter such as X, can take values from a set of distinct outcomes. For example, the number of heads when flipping three coins can be 0, 1, 2, or 3. These values are countable and separated.

离散型随机变量通常用大写字母X表示，可以从一组分离的结果中取值。例如，投掷三枚硬币时出现正面的次数可以是0、1、2或3。这些值是可数的且彼此分离。

In contrast, a continuous random variable can take any value within an interval, such as height or time. The focus of this revision is on discrete cases.

相比之下，连续型随机变量可以在一个区间内取任意值，比如身高或时间。本次复习的重点是离散情况。

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2. Probability Distribution of a Discrete Random Variable | 离散型随机变量的概率分布

To fully describe a discrete random variable, we list its possible values along with their corresponding probabilities. This can be done using a probability distribution table. For instance, the roll of a fair six-sided die gives each outcome probability 1/6.

为了完整描述一个离散型随机变量，我们列出它的所有可能值及相应的概率。这可以使用概率分布表来完成。例如，投掷一个公平的六面骰子，每个结果的概率为1/6。

wo rows: one for values x and one for P(X = x). The table must satisfy certain conditions.

概率分布表通常有两行：一行是x的值，另一行是P(X = x)。该表必须满足一些条件。

x	1	2	3	4	5	6
P(X = x)	1/6	1/6	1/6	1/6	1/6	1/6

This table shows a simple discrete uniform distribution, a special case we will revisit later.

该表格展示了一个简单的离散均匀分布，一个稍后会回顾的特例。

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3. Conditions for a Valid Probability Distribution | 有效概率分布的条件

, 0 ≤ P(X = x) ≤ 1 for all x. Secondly, the sum of all probabilities must equal exactly 1.

一个函数要成为离散型随机变量的概率分布，必须满足两个条件：第一，每个概率必须在0到1之间（含0和1），即对于所有x，0 ≤ P(X = x) ≤ 1。第二，所有概率之和必须恰好等于1。

These conditions ensure that the probabilities are consistent and that the variable must take one of the listed values. If either condition fails, the distribution is not valid.

这些条件确保了概率的一致性，并且变量一定会取所列值中的一个。如果任一条件不满足，分布就是无效的。

In exam questions, you might be asked to find a missing probability or to verify whether a given table represents a probability distribution.

在考试中，可能会要求你找出缺失的概率，或验证给定的表格是否表示一个概率分布。

Σ P(X = x) = 1

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4. Expected Value E(X) – The Mean | 期望值E(X)——均值

The expected value, or mean, of a discrete random variable X is a measure of its central tendency. It is calculated by multiplying each possible value by its probability and summing the results.

离散型随机变量X的期望值或均值是衡量其集中趋势的指标。计算方法是将每个可能值乘以其概率，然后求和。

E(X) = μ = Σ x · P(X = x)

) + ... + 6×(1/6) = 3.5. Note that the expected value need not be a possible value itself.

例如，如果X表示投掷一个公平骰子的结果，E(X) = 1×(1/6) + 2×(1/6) + ... + 6×(1/6) = 3.5。请注意，期望值本身不一定是可能取到的值。

The expected value is the long-run average if the experiment is repeated many times.

期望值是在多次重复实验下的长期平均值。

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5. Variance and Standard Deviation | 方差与标准差

tion from the mean. An equivalent and often easier computational formula is Var(X) = E(X²) – [E(X)]².

方差衡量分布围绕均值的离散程度。对于离散型随机变量X，方差Var(X)定义为与均值之差的平方的期望值。一个等价且通常更容易计算的形式是Var(X) = E(X²) – [E(X)]²。

Var(X) = σ² = Σ (x – μ)² P(X = x) = Σ x² P(X = x) – μ²

P(X = x). The standard deviation is the positive square root of the variance: σ = √Var(X).

其中E(X²) = Σ x² P(X = x)。标准差是方差的正平方根：σ = √Var(X)。

air die example, E(X²) = 1²×(1/6)+2²×(1/6)+...+6²×(1/6) = 91/6 ≈ 15.1667. So Var(X) = 91/6 – (3.5)² = 35/12 ≈ 2.9167, and σ ≈ 1.7078.

以公平骰子为例，E(X²) = 1²×(1/6)+2²×(1/6)+...+6²×(1/6) = 91/6 ≈ 15.1667。因此Var(X) = 91/6 – (3.5)² = 35/12 ≈ 2.9167，σ ≈ 1.7078。

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6. Linear Transformations of a Random Variable | 随机变量的线性变换

= aX + b, where a and b are constants, its mean and variance change in a predictable way.

当随机变量X经过线性变换Y = aX + b（其中a和b为常数）后，其均值和方差会以可预测的方式变化。

E(aX + b) = a E(X) + b

Var(aX + b) = a² Var(X)

The expected value is shifted by the constant b and scaled by a. The variance is scaled by a² because variance measures squared deviations; adding a constant b does not affect spread.

期望值被常数b平移并被a缩放。方差被a²缩放，因为方差衡量偏差的平方；加上常数b不影响离散程度。

ngs X have E(X)=5 and Var(X)=16, and the casino charges a fee such that net gain is Y = 2X – 3, then E(Y) = 2×5 – 3 = 7, and Var(Y) = 2² × 16 = 64.

例如，如果赌博奖金X的期望值E(X)=5，方差Var(X)=16，赌场收取费用后净收益为Y = 2X – 3，则E(Y) = 2×5 – 3 = 7，Var(Y) = 2² × 16 = 64。

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7. The Discrete Uniform Distribution | 离散均匀分布

probability 1/n. An example is a fair die (n=6) or a spinner with n equally sized sectors.

一种特殊的离散分布是均匀分布，其中n种可能结果出现的概率都相等，为1/n。例如公平骰子（n=6）或具有n个等分扇区的转盘。

If X follows a discrete uniform distribution on the consecutive integer values {1, 2, ..., n}, then:

若X服从连续整数{1, 2, ..., n}上的离散均匀分布，则：

E(X) = (n + 1) / 2

Var(X) = (n² – 1) / 12

s without tedious summation. For n=6, E(X)=3.5 and Var(X)=35/12, matching earlier results.

这些公式可以在无需繁琐求和的情况下快速计算。当n=6时，E(X)=3.5，Var(X)=35/12，与之前的结果吻合。

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8. Introduction to the Binomial Distribution | 二项分布简介

One of the most frequently applied discrete distributions is the binomial distribution. It models the number of successes in a fixed number n of independent Bernoulli trials, each with the same success probability p.

应用最广泛的离散分布之一是二项分布。它描述了在固定次数n的独立伯努利试验

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

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