Finding μ and σ | 求均值与标准差

📚 Finding μ and σ | 求均值与标准差

In Edexcel A-Level Mathematics, a common type of question involves finding the mean (μ) and standard deviation (σ) of a normal distribution when certain probabilities are given. This skill requires a solid understanding of the standard normal distribution, z-scores, and the inverse normal function.

在Edexcel A-Level数学中，常见题型是根据给定的概率求正态分布的均值(μ)和标准差(σ)。这需要牢固掌握标准正态分布、Z分数以及逆正态函数。

1. The Normal Distribution Recap | 正态分布回顾

The normal distribution is defined by two parameters: μ (mean) and σ (standard deviation). The probability that the random variable X falls within a certain interval is represented by the area under the bell-shaped curve. All normal distributions are symmetric about the mean.

正态分布由两个参数定义：μ（均值）和σ（标准差）。随机变量X落在某区间的概率由钟形曲线下的面积表示。所有正态分布关于均值对称。

For any X ~ N(μ, σ²), we can transform it to a standard normal variable Z ~ N(0, 1) using the formula shown below.

对于任何X ~ N(μ, σ²)，我们可以通过下面的公式将其转化为标准正态变量Z ~ N(0, 1)。

Z = (X − μ) / σ

This standardisation allows us to use tables or calculators to find probabilities or z-values corresponding to given probabilities.

这种标准化使我们能够使用表格或计算器找出概率或与给定概率对应的z值。

2. Standardising: The Z-Formula | 标准化：Z公式

The z-score tells us how many standard deviations an observed value is from the mean. If a probability is given, such as P(X < a) = p, we can find the corresponding z-value, z = Φ⁻¹(p), where Φ is the cumulative distribution function of the standard normal distribution. Then we set up the equation: z = (a − μ) / σ.

z分数表示观测值距离均值多少个标准差。如果给定了概率，例如P(X < a)=p，我们可以找到对应的z值z = Φ⁻¹(p)，其中Φ是标准正态分布的累积分布函数。然后建立方程：z = (a − μ) / σ。

If the probability is right-tail, e.g. P(X > a) = p, we convert it to a left-tail probability: P(X < a) = 1 − p, then find the appropriate z.

如果概率是右尾的

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