20.4.7 Normal distributions
The probability density function for a normal distribution.
The density function of the normal distribution with mean µ and
standard deviation σ at the point x is
normald(µ,σ,x)= | | e(x−µ)2/2.
(4) |
The normald
(or loi_normal) command finds the
value of this density function.
-
normald (or loi_normal) command takes two
optional arguments and one mandatory argument:
-
Optionally, µ and σ, the mean and standard
deviation. (By default, µ=0 and σ=1, giving the
standard normal distribution.)
- x, a real number.
- normald(⟨µ,σ,⟩ x) returns the value of the
normal density function with parameter µ and standard deviation
σ at the value x, given in (4).
Examples
The cumulative distribution function for normal distribution.
The normal_cdf
(or normald_cdf)
command computes the cumulative distribution function for the normal distribution.
-
normal_cdf (or normald_cdf) takes three
optional arguments and one mandatory argument:
-
Optionally, µ and σ, the mean and standard
deviation. (By default, µ=0 and σ=1, giving the
standard normal distribution.)
- x, a real number.
- Optionally, y, a real number.
- normal_cdf(⟨µ,σ,⟩ x) returns
Prob(X ≤ x) for the normal distribution with mean µ
and standard deviation σ.
- normal_cdf(⟨µ,σ,⟩ x,y) returns
Prob(x ≤ X ≤ y).
Examples
The inverse distribution function for normal distribution.
The normal_icdf
(or normald_icdf)
command computes the inverse distribution for the normal distribution.
-
normal_icdf (or normald_icdf) takes two
optional arguments and one mandatory argument:
-
Optionally, µ and σ, the mean and standard
deviation. (By default, µ=0 and σ=1, giving the
standard normal distribution.)
- h, a real number.
- normal_icdf(⟨µ,σ,⟩ h) returns the inverse
distribution for the normal distribution with mean µ and
standard deviation σ; namely, the value of x for which
Prob(X ≤ x)=h.
Examples
The upper tail cumulative function for normal distributions.
The UTPN
(the Upper Tail Probability-Normal distribution)
computes Prob(X > x) for a normal distribution.
-
UTPN takes two
optional arguments and one mandatory argument:
- UTPN(⟨µ,σ2,⟩ x) returns
Prob(X > x), for the normal distribution with mean
µ and variance σ2.
Examples