摘要：随机变量（Random Variable）是一个从样本空间到实数的函数，它将每一个可能的结果映射到一个实数值。随机变量通常用大写字母表示，如X。
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思维导图
Mind mapping
随机变量及其分布的基本概念：
random Variable
随机变量（Random Variable）是一个从样本空间到实数的函数，它将每一个可能的结果映射到一个实数值。随机变量通常用大写字母表示，如X。
A random variable is a function from the sample space to the set of real numbers, which maps each possible outcome to a real number. Random variables are usually represented by uppercase letters, such as X.
随机变量可以分为以下两大类：
Random variables can be divided into the following two major categories:
离散随机变量：只能取有限个或可数无限个值。例如，掷骰子的结果就是一个离散随机变量。
Discrete random variable: It can take a finite number or a countably infinite number of values. For example, the result of rolling a die is a discrete random variable.
连续随机变量：可以取某个区间内的任何值。例如，某城市一天的气温就是一个连续随机变量。
Continuous random variable: It can take any value within a certain interval. For example, the daily temperature in a city is a continuous random variable.
几何分布：在n次伯努利试验中，首次成功发生在第k次的概率分布。
Geometric Distribution: The probability distribution of the first success occurring on the kth trial in n Bernoulli trials.
泊松分布：在连续时间内发生某一事件的次数的概率分布，适用于事件发生率固定且独立的情况。
Poisson Distribution: The probability distribution of the number of times an event occurs in a continuous time interval, suitable for situations where the event occurrence rate is fixed and independent.
均匀分布：在某个区间内取值概率相同的分布。
Uniform Distribution: A distribution where the probability of taking values is the same within a certain interval.
正态分布：自然界中最常见的连续概率分布，呈对称的钟形曲线。
Normal Distribution: The most common continuous probability distribution in nature, characterized by a symmetric bell-shaped curve.
了解随机变量及其分布是进行数据分析、推断和决策的基础。在实际应用中，根据具体情况选择合适的随机变量模型，可以帮助我们更好地理解和预测现实世界中的随机现象。
Understanding random variables and their distributions is the basis for data analysis, inference, and decision-making. In practical applications, choosing the appropriate random variable model according to the specific situation can help us better understand and predict random phenomena in the real world.
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