6.2:

6.2: 随机变量

Random Variables

JoVE Core
Statistics

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JoVE Core Statistics

Random Variables

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6.1: Probability in Statistics

6.3: Probability Distributions

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01:09 min

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April 30, 2023

Overview

随机变量是指示过程结果的单个数值。随机变量的概念是概率论的基础，由俄罗斯数学家帕夫努蒂·切比雪夫（Pafnuty Chebyshev）在 19 世纪中叶提出。

大写字母（如 X 或 Y）表示随机变量。小写字母（如 x 或 y）表示随机变量的值。如果 X 是随机变量，则 X 是用单词写的，而 x 是数字。

例如，设 X = 当您抛出三个公平的硬币时获得的正面数量。抛出三个公平硬币的样本空间是 TTT;THH;HTH;HHT;HTT;THT;TTH;HHH.然后，x = 0， 1， 2， 3。X 是文字，x 是一个数字。请注意，在此示例中，x 值是可计数的结果。

随机变量可以有两种类型:离散随机变量和连续随机变量。

离散随机变量是具有有限数量的变量。换句话说，随机变量是一个可数数。例如，骰子上的数字 1、2、3、4、5 和 6 是离散随机变量。

连续随机变量是具有连续刻度的值而没有间隙或中断的变量。连续随机变量表示为十进制值。一个例子是学生的身高 – 1.83 m。

本文改编自 Openstax，统计学导论，部分。4 引言

Transcript

考虑掷骰子 30 次。在每次试验中，结果可以是 1 到 6 之间的任何结果。如果一个人在 30 次中出现 6 次，则它的概率是 30 次中有 6 次，依此类推。

这些结果中的每一个（称为随机变量）都有一个随机确定的数值。它表示实验的所有可能结果。

小写字母 x 表示随机变量的数值。

随机变量可以是离散的，也可以是连续的。

离散随机变量可以与有限或无限的计数过程相关联。例如，一只母鸡可能会产一个鸡蛋、两个鸡蛋或更多，但不能产 1.27 个鸡蛋。

相反，连续随机变量具有无限多个值，这些值可以与连续尺度上没有间隙或中断的测量值相关联。

例如，在一天内，一头奶牛可能会生产 0 到 20 升牛奶，以连续刻度测量。

Key Terms and definitions

Random Variable - A single numeric outcome of a procedure, influenced by chance.
Discrete Random Variable - A countable number or finite quantity, like die faces.
Continuous Random Variable - Infinite possible values from a continuous scale, e.g., student height.
Pafnuty Chebyshev - The mathematician who introduced the concept of random variables.
Probability Theory - The academic field where the concept of random variables is fundamental.

Learning Objectives

Define Random Variable – This indicates a single numerical outcome of a process (e.g., dice roll).
Contrast Discrete vs Continuous Random Variables – Understand how they differ in terms of value possibilities (e.g., die faces vs student height).
Explore Example –Find out how the number of heads in coin tosses fits into this context (e.g., possible outcomes scenario).
Explain Chebyshev's contribution – Understand who introduced the concept of random variables and its significance.
Apply in Statistics – Grasp how random variables play a key role in probability theory.

Questions that this video will help you answer

What is a random variable and how is it determined by chance?
How do discrete and continuous random variables differ?
Who is Pafnuty Chebyshev and what was his contribution to probability theory?

This video is also useful for

Students – Understand how the concept of random variables supports learning in statistics and probability.
Educators – Provides a clear framework for teaching the concept of random variables and its types.
Researchers – Importance of random variables in developing statistical models and probability theory.
Science Enthusiasts – Explores the idea of randomness and variability in a scientific context.

Tags

随机变量概率论帕夫努蒂·切比雪夫离散随机变量连续随机变量样本空间可计数结果有限数量高度测量数值