normL: Normal - Likelihood

In jeksterslabds/jeksterslabRdist: Jekster's Lab Distribution Utilities

Description

Calculates the likelihood of X following a normal distribution.

Usage

normL(mu, sigma, x)

Arguments

mu	Numeric. Location parameter mean μ.
sigma	Numeric. Positive number. Scale parameter standard deviation σ = √{σ^2}.
x	Numeric vector. Values of the random variable X.

Details

The likelihood function for the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is given by

\mathcal{L} ≤ft( μ, σ^2 \mid x \right) = ∏_{i = 1}^{n} ≤ft\{ \frac{1}{σ √{2 π}} \exp ≤ft[ - \frac{1}{2} ≤ft( \frac{x_i - μ}{σ} \right)^2 \right] \right\} %(\#eq:dist-normL-1)

or

\mathcal{L} ≤ft( μ, σ^2 \mid x \right) = ∏_{i = 1}^{n} ≤ft\{ \frac{1}{√{2 π σ^2}} \exp ≤ft[ - \frac{ ≤ft( x_i - μ \right)^2} {2 σ^2} \right] \right\} \\ = ≤ft( \frac{1}{√{2 π σ^2}} \right)^n \exp ≤ft[ - \frac{1}{2 σ^2} ∑_{i = 1}^{n} ≤ft( x_i - μ \right)^2 \right] %(\#eq:dist-normL-2)

with independent and identically distributed sample data x \in \mathbf{R}, μ is the location parameter mean being estimated ≤ft( μ \in \mathbf{R} \right), and σ^2 is the scale parameter variance being estimated ≤ft( σ^2 > 0 \right).

Author(s)

Ivan Jacob Agaloos Pesigan

References

Wikipedia: Normal Distribution

Wikipedia: IID

Wikipedia: Likelihood Function

See Also

Other normal likelihood functions: norm2ll(), normll(), normobj(), normpdf()

jeksterslabds/jeksterslabRdist documentation built on Aug. 9, 2020, 7:33 a.m.