Probability density function The red line is the standard normal distribution |
Cumulative distribution function Colors match the image above |
Notation |
![{\displaystyle {\mathcal {N}}(\mu ,\,\sigma ^{2})}](//wikimedia.org/api/rest_v1/media/math/render/svg/34ce0ec18437bd01cec5958c39d732d4ddaf7530) |
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Parameters |
μ ∈ R — mean (location) σ2 > 0 — variance (squared scale) |
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Support |
x ∈ R |
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PDF |
![{\displaystyle {\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\,e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1a9287a082350af2fe84ea67da609e32f8591528) |
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CDF |
![{\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sqrt {2\sigma ^{2}}}}\right)\right]}](//wikimedia.org/api/rest_v1/media/math/render/svg/04670b14acb4ddb796469f3812ead9d9cccec275) |
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Mean |
μ |
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Median |
μ |
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Mode |
μ |
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Variance |
σ2 |
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Skewness |
0 |
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Excess kurtosis |
0 |
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Entropy |
![{\displaystyle {\frac {1}{2}}\ln(2\pi e\,\sigma ^{2})}](//wikimedia.org/api/rest_v1/media/math/render/svg/283bb3954fd82cdd6a67108e7561d98593e459e4) |
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MGF |
![{\displaystyle \exp\{\mu t+{\frac {1}{2}}\sigma ^{2}t^{2}\}}](//wikimedia.org/api/rest_v1/media/math/render/svg/c7d888a0cb5ee432d2d384fafb285d46152aa8f5) |
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CF |
![{\displaystyle \exp\{i\mu t-{\frac {1}{2}}\sigma ^{2}t^{2}\}}](//wikimedia.org/api/rest_v1/media/math/render/svg/70ca5afd79af42578675e292e2b25d578133d450) |
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Fisher information |
![{\displaystyle {\begin{pmatrix}1/\sigma ^{2}&0\\0&1/(2\sigma ^{4})\end{pmatrix}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/262dc78a347dc277885993a5cc349a16547ae366) |
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