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

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:
  • Optionally, µ and σ², the mean variance deviation. (By default, µ=0 and σ²=1, giving the standard normal distribution.)

    Note that, unlike normald and normal_cdf, the UTPN takes the variance and not the standard deviation.

  • x, a real number.
• UTPN(⟨µ,σ²,⟩ x) returns Prob(X > x), for the normal distribution with mean µ and variance σ².

Examples