Se notează cu: N(μ,σ), unde μ și σ sunt parametrii din funcția de distribuție care va fi descrisă în continuare.
f
(
x
)
=
1
σ
2
π
e
−
(
x
−
μ
)
2
2
σ
2
{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}}
M
(
X
)
{\displaystyle M(X)\,}
=
∫
−
∞
∞
x
.
f
(
x
)
d
x
{\displaystyle \int _{-\infty }^{\infty }x.f(x)\,dx}
=
∫
−
∞
∞
x
.
1
σ
2
π
e
−
(
x
−
μ
)
2
2
σ
2
d
x
{\displaystyle \int _{-\infty }^{\infty }x.{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,dx}
=
μ
{\displaystyle \mu }
σ
2
(
X
)
=
∫
−
∞
∞
(
x
−
M
(
X
)
)
2
.
f
(
x
)
d
x
{\displaystyle \sigma ^{2}(X)=\int _{-\infty }^{\infty }(x-M(X))^{2}.f(x)\,dx}
=
∫
−
∞
∞
(
x
−
M
(
X
)
)
2
.
1
σ
2
π
e
−
(
x
−
μ
)
2
2
σ
2
d
x
{\displaystyle \int _{-\infty }^{\infty }(x-M(X))^{2}.{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,dx}
=
∫
−
∞
∞
(
x
−
μ
)
2
.
1
σ
2
π
e
−
(
x
−
μ
)
2
2
σ
2
d
x
{\displaystyle \int _{-\infty }^{\infty }(x-\mu )^{2}.{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\,dx}
=
σ
2
{\displaystyle \sigma ^{2}}
H
[
f
]
=
−
∫
−
∞
∞
f
(
x
)
ln
(
f
(
x
)
)
d
x
{\displaystyle H[f]=-\int \limits _{-\infty }^{\infty }f(x)\ln(f(x))\,dx}
=
−
∫
−
∞
∞
1
σ
2
π
e
−
(
x
−
μ
)
2
2
σ
2
ln
(
1
σ
2
π
e
−
(
x
−
μ
)
2
2
σ
2
)
d
x
{\displaystyle -\int \limits _{-\infty }^{\infty }{\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}}\ln({\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {(x-\mu )^{2}}{2\sigma ^{2}}}})\,dx}
=
ln
(
σ
2
π
e
)
{\displaystyle \ln \left(\sigma {\sqrt {2\,\pi \,e}}\right)\!}
Funcția de repartiție cumulativă este funcția
F
(
t
)
{\displaystyle F(t)}
=
∫
−
∞
t
f
(
x
)
d
x
{\displaystyle \int _{-\infty }^{t}f(x)\,dx}
=
1
σ
.
