V2EX q-function

Q-Function

定义 Definition

Q-function(Q函数)是概率与通信领域常用的函数,通常指标准正态分布的上尾概率
\(Q(x)=P(Z>x)\),其中 \(Z\sim N(0,1)\)。它用来表示“超过某个阈值的概率”,在误码率(BER)分析等场景非常常见。(在某些语境中也可能指其他“Q函数”,但最常见的是正态分布的这个定义。)

发音 Pronunciation (IPA)

/kju fkn/

例句 Examples

The Q-function gives the probability that a standard normal variable exceeds x.
Q函数给出标准正态变量超过 x 的概率。

In digital communication theory, the bit error rate is often expressed using the Q-function as a function of SNR.
在数字通信理论中,比特误码率常常用Q函数表示,并写成信噪比(SNR)的函数。

词源 Etymology

“Q-function”中的 Q 多被理解为一个约定俗成的符号名,用于表示正态分布的“尾部(tail)概率”函数;该记号在统计学与工程(尤其通信工程)文献中长期沿用。它与互补误差函数有紧密关系:
\(Q(x)=\tfrac12\,\mathrm{erfc}\!\left(\tfrac{x}{\sqrt2}\right)\)。

相关词 Related Words

文献作品 Literary Works

  • Digital Communications(John G. Proakis):用Q函数推导与表达误码率等性能指标
  • Communication Systems(Simon Haykin):在噪声环境下的检测与误差概率分析中使用Q函数
  • Probability, Random Variables, and Stochastic Processes(Papoulis & Pillai):讨论正态分布相关概率表达时涉及Q函数/尾概率表示法
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