gaussian

func (*Distribution) CDF ¶

func (d *Distribution) CDF(x float64) float64

CDF calculates the cumulative distribution function (CDF) of the Gaussian distribution.

The CDF of a normal (Gaussian) distribution provides the probability of a random value (X) drawn from the distribution will be less than or equal to a specific value x.

The CDF of a normal distribution with mean μ and standard deviation σ is typically denoted as: F(x) = P[X ≤ x]

F(x) = 0.5 * (1 + erf((x - μ) / (σ * √2))) = 0.5 * erfc(-(x - μ) / (σ * √2))

Where:

• x: The value at which to calculate the CDF
• σ(sigma): The standard deviation of the distribution.
• μ(mu): The mean of the distribution.
• erf(z): The error function
• erfc(z): The complementary error function(math.Erfc) , erfc(z) = 1 - erf(z) (math.Erf)

func (*Distribution) Exponent ¶

func (d *Distribution) Exponent(x float64) float64

Exponent calculates the core Exponent term in the Gaussian distribution formula.

This function is core to calculating the probability density and other related values.

The core term of a Gaussian distribution(or normal distribution) is defined as:

e^(-(x - μ)^2 / (2σ^2))

Where:

• x: The value at which to evaluate the Gaussian function.
• μ(mu): The mean of the distribution (the centre of the bell curve).
• σ(sigma): The standard deviation of the distribution (controls the spread of the curve).
• σ^2: The variance of the distribution
• e - Mathematical constant math.E

This Exponent term determines the relative likelihood of a value `x` within the Gaussian distribution.

• The closer `x` to the mean `μ`, the closer the Exponent term gets to 1, indicating higher probability density.
• As `x` moves away from the mean `μ`, the Exponent term decreases, indicating lower probability density.
• The standard deviation `σ` controls the spread: Smaller standard deviation leads to sharper curves with probability density concentrated around the mean.

func (*Distribution) PDF ¶

func (d *Distribution) PDF(x float64) float64

PDF calculates the probability density function (PDF) at a given value.

PDF of a Gaussian distribution is a bell-shaped curve, that describes the relative likelihood of a random variable taking on a specific value. Formula:

f(x) = (1 / (σ * √(2π))) * e^(-(x - μ)^2 / (2σ^2))

where:

• x: The value for which we want to calculate the probability density.
• μ(mu): The mean of the distribution (the centre of the bell curve).
• σ(sigma): The standard deviation of the distribution (controls the spread of the curve).
• σ^2: The variance of the distribution
• π - Mathematical constant math.Pi
• e - Mathematical constant math.E