Search
Search Icon Icon to open search 正态分布_高斯分布_Normal_Distribution-Gaussian_Distribution Last updated
Sep 16, 2021
Edit Source
# 正态分布2021-09-16
Tags: #Math/Statistics
# 概率密度函数正态分布, 概率密度函数:
f ( x ) = 1 σ 2 π e − 1 2 ( x − μ σ ) 2 f(x)=\frac{1}{\sigma \sqrt{2 \pi}} e^{\Large -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}} f ( x ) = σ 2 π 1 e − 2 1 ( σ x − μ ) 2
or
f ( x ) = 1 σ 2 π exp ( − 1 2 ( x − μ σ ) 2 ) f(x)=\frac{1}{\sigma \sqrt{2 \pi}} \exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right) f ( x ) = σ 2 π 1 exp ( − 2 1 ( σ x − μ ) 2 )
# 重要性质Mean ( μ ) (\mu) ( μ ) and standard deviation ( σ ) (\sigma) ( σ )
\begin{aligned}
&\mu=E(X)=\int_{-\infty}^{\infty} x p(x) d x \\ &\sigma^{2}=E\left{(X-\mu)^{2}\right}=\int_{-\infty}^{\infty}(x-\mu)^{2} p(x) d x
\end{aligned}
Probability within any particular number of standard deviations of μ \mu μ
p μ − k σ ≤ x ≤ μ + k σ = ∫ μ − k σ μ + k σ 1 2 π σ exp [ − ( x − μ ) 2 2 σ 2 ] d x = 1 2 π ∫ − k k exp [ − y 2 2 ] d y
\begin{aligned}
p{\mu-k \sigma \leq x \leq \mu+k \sigma} &=\int_{\mu-k \sigma}^{\mu+k \sigma} \frac{1}{\sqrt{2 \pi} \sigma} \exp \left[-\frac{(x-\mu)^{2}}{2 \sigma^{2}}\right] d x \\ &=\frac{1}{\sqrt{2 \pi}} \int_{-k}^{k} \exp \left[-\frac{y^{2}}{2}\right] d y
\end{aligned}
p μ − kσ ≤ x ≤ μ + kσ = ∫ μ − kσ μ + kσ 2 π σ 1 exp [ − 2 σ 2 ( x − μ ) 2 ] d x = 2 π 1 ∫ − k k exp [ − 2 y 2 ] d y
线性变换后:
如果 X ∼ N ( μ , σ 2 ) X \sim N\left(\mu,\sigma^{2}\right) X ∼ N ( μ , σ 2 ) 且 a , b a, b a , b 是实数, 那么
a X + b ∼ N ( a μ + b , ( a σ ) 2 ) a X+b \sim N\left(a \mu+b,(a \sigma)^{2}\right) a X + b ∼ N ( a μ + b , ( aσ ) 2 )
正态分布的和还是正态分布
X ∼ N ( μ X , σ X 2 ) Y ∼ N ( μ Y , σ Y 2 ) Z = X + Y \begin{aligned}
&X \sim N\left(\mu_{X}, \sigma_{X}^{2}\right) \\ &Y \sim N\left(\mu_{Y}, \sigma_{Y}^{2}\right) \\ &Z=X+Y
\end{aligned} X ∼ N ( μ X , σ X 2 ) Y ∼ N ( μ Y , σ Y 2 ) Z = X + Y
则
Z ∼ N ( μ X + μ Y , σ X 2 + σ Y 2 ) Z \sim N\left(\mu_{X}+\mu_{Y}, \sigma_{X}^{2}+\sigma_{Y}^{2}\right) Z ∼ N ( μ X + μ Y , σ X 2 + σ Y 2 )
# 记忆公式注意σ \sigma σ 在根号外面 指数是负的(x = μ x=\mu x = μ 的时候等于0, 同时取得最大值) # 与拉普拉斯分布的联系拉普拉斯分布与高斯分布的联系_Relation_of_Laplace_distribution _and_Gaussian_distribution
拉普拉斯分布与高斯分布的联系_Relation_of_Laplace_distribution _and_Gaussian_distribution
Gaussian distribution, Laplace distribution: The Relation
2021-07-31
Tags: #Math/Statistics #GaussianDistribution #LaplaceDistribution
拉普拉斯分布, 概率密度函数:
Look at the formula for the PDF in the infobox...
11/19/2023
# Higher Dimensionsp x = 1 ( 2 π ) n / 2 ∣ C ∣ 1 / 2 exp [ − 1 2 ( x − μ ) T C − 1 ( x − μ ) ] x = [ x 1 ⋯ x n ] μ = [ μ 1 ⋯ μ n ] C = [ σ 11 2 … σ 1 n 2 ⋯ … σ m 1 2 … σ m n 2 ] \begin{aligned}
&p{x}=\frac{1}{(\sqrt{2 \pi})^{n / 2}|C|^{1 / 2}} \exp \left[-\frac{1}{2}(x-\mu)^{T} C^{-1}(x-\mu)\right] \\ &x=\left[\begin{array}{c}
x_{1} \\ \cdots \\ x_{n}
\end{array}\right] \quad \mu=\left[\begin{array}{c}
\mu_{1} \\ \cdots \\ \mu_{n}
\end{array}\right] \quad C=\left[\begin{array}{ccc}
\sigma_{11}^{2} & \ldots & \sigma_{1 n}^{2} \\ \cdots & & \ldots \\ \sigma_{m 1}^{2} & \ldots & \sigma_{m n}^{2}
\end{array}\right]
\end{aligned} p x = ( 2 π ) n /2 ∣ C ∣ 1/2 1 exp [ − 2 1 ( x − μ ) T C − 1 ( x − μ ) ] x = ⎣ ⎡ x 1 ⋯ x n ⎦ ⎤ μ = ⎣ ⎡ μ 1 ⋯ μ n ⎦ ⎤ C = ⎣ ⎡ σ 11 2 ⋯ σ m 1 2 … … σ 1 n 2 … σ mn 2 ⎦ ⎤