Gaussian(Normal) Distribution
The probability density function (PDF) is
$$ f(x\,|\,\mu,\,\sigma) = \frac{1}{\sigma\sqrt{2\pi}}\,\exp\!\Bigg( -\,\frac{(x-\mu)^2}{2\sigma^2} \Bigg) $$
Current parameters: $\mu=0.00,\ \sigma=1.00$
Controls
68% Interval[μ−σ, μ+σ]
95% Interval[μ−2σ, μ+2σ]
99.7% Interval[μ−3σ, μ+3σ]
Standard Identities
Standardization: $$ Z = \frac{X-\mu}{\sigma} \sim \mathcal{N}(0,1) $$
CDF: $$ F(x) = \mathbb{P}(X\le x) = \tfrac{1}{2}\bigg[1+\operatorname{erf}\!\Big(\tfrac{x-\mu}{\sigma\sqrt{2}}\Big)\bigg] $$
Moments: $$ \mathbb{E}[X]=\mu,\quad \operatorname{Var}(X)=\sigma^2 $$
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