Normal distribution

Normal distribution/ Gaussian distribution Sumarize Probability density function The red curve is the standard normal distribution Cumulative distribution function Notation Parameters μ ∈ R — mean σ2 > 0 — variance Support（定义域） x ∈ R (-∞,+∞) pdf(概率密度函数) CDF（累计分布函数） Mean μ Median μ Mode（众数） μ Variance Skewness（偏度） 0 Kurtosis（峰度） 31 Standard normal distribution If μ = 0 and σ = 1, the distribution is called the standard normal distribution or the unit normal distribution, and a random variable with that distribution is a standard normal deviate. 2 1 关于偏度为 0和峰度为 3的原因见下文Moments部分 2 积分和为 1的证明：先令 I=∫f(x)dx，再将一重积分转换为二重积分，再通过极坐标变换得出 Properties ·If X~N(μ, ), for any real numbers a and b, then: aX + b ~N(aμ+b, a2 ). ·If X ~N(μX, X) and Y~N(μY, Y) are two independent normal random variables, then： U = X + Y ~N(μX+μY, X+ Y). 3 V = X - Y ~N(μX-μY, X+ Y). 4 ·If X and Y are jointly normal and uncorrelated, then they are independent. 5 Moments If X (mean is 0) has a normal distribution, these moments exist and are finite for any p whose real part is greater than −1. For any non-negative integer p, the plain central moments are 6 Here n!! denotes the double factorial, that is the product of every odd number from n to 1. 3 可由定义及卷积公式证明，下同 4 若 X与 Y的方差相等，则 U与 V两者相互独立 5 The requirement that X and Y should be jointly normal is essential, without it the property does not hold. For non-normal random variables uncorrelatedness does not imply independence. 6 当 p是奇数时，Ex^p=∫x^p*f(x)dx, 奇函数乘以偶函数得奇函数，奇函数在 R上的定积分为 0 The central absolute moments coincide with plain moments for all even orders, but are nonzero for odd orders. For any non-negative integer p, Order Non-central moment Central moment 1 μ 0 2 μ2 + σ2 σ 2 3 μ3 + 3μσ2 0 4 μ4 + 6μ2σ2 + 3σ4 3σ 4 5 μ5 + 10μ3σ2 + 15μσ4 0 6 μ6 + 15μ4σ2 + 45μ2σ4 + 15σ6 15σ 6 7 μ7 + 21μ5σ2 + 105μ3σ4 + 105μσ6 0 8 μ8 + 28μ6σ2 + 210μ4σ4 + 420μ2σ6 + 105σ8 105σ 8 Combination of two or more independent random variables  If X1 and X2 are two independent standard normal random variables with mean 0 and variance 1, then their sum and difference is distributed normally with mean zero and variance two:X1 ± X2 ∼ N(0, 2).  If X1, X2, …, Xn are independent standard normal random variables, then the sum of their squares has the chi-squared distribution with n degrees of freedom .  If X1, X2, …, Xn are independent normally distributed random variables with means μ and variances σ 2 , then their sample mean is independent from the sample standard deviation, which can be demonstrated using Basu's theorem or Cochran's theorem. The ratio of these two quantities will have the Student's t-distribution with n − 1 degrees of freedom:  If X1, …, Xn, Y1, …, Ym are independent standard normal random variables, then the ratio of their normalized sums of squares will have the F-distribution with (n, m) degrees of freedom: