normal distribution

URI: https://gptkb.org/entity/normal_distribution
GPTKB entity
AI-created image of normal distribution
AI-created image
Statements (101)
Predicate Object
gptkbp:instanceOf gptkb:organization
probability_distribution
gptkbp:affiliatedWith continuous_distribution
exponential_family
location-scale_family
gptkbp:alsoKnownAs gptkb:Gaussian_distribution
Gaussian_distribution
gptkbp:central_limit_theorem limit_distribution
gptkbp:centralLimitTheorem limit distribution
gptkbp:characteristic exp(iμt - 0.5σ²t²)
gptkbp:characteristic_function exp(i*t*mu - 0.5*sigma^2*t^2)
gptkbp:continuity true
gptkbp:cumulative_distribution_function 0.5*(1 + erf((x-mu)/(sigma*sqrt(2))))
gptkbp:cumulativeDistributionFunction (1/2)[1 + erf((x-μ)/(σ√2))]
gptkbp:entropy 0.5*ln(2πeσ²)
0.5*ln(2*pi*e*sigma^2)
gptkbp:first_moment mu
gptkbp:fourth_moment mu^4 + 6*mu^2*sigma^2 + 3*sigma^4
gptkbp:hasSpecialCase gptkb:exponential_family
gptkb:location-scale_family
https://www.w3.org/2000/01/rdf-schema#label normal distribution
gptkbp:is_limit_of_binomial true
gptkbp:is_limit_of_poisson true
gptkbp:is_symmetric true
gptkbp:is_unimodal true
gptkbp:isConjugatePriorFor mean of normal distribution
variance of normal distribution
gptkbp:isEllipticalDistribution true
gptkbp:isInfinitelyDivisible true
gptkbp:isMaximumEntropyDistribution true
gptkbp:isSelfDecomposable true
gptkbp:isStableDistribution true
gptkbp:isUnimodal true
gptkbp:kurtosis 3
gptkbp:limitation gptkb:binomial_distribution_(as_n→∞,_p→0,_np=μ)
Poisson distribution (as λ→∞)
sum of independent random variables (central limit theorem)
gptkbp:maximum_entropy_distribution true
gptkbp:maximumLikelihoodEstimator sample mean and sample variance
gptkbp:meaning μ
mu
gptkbp:medium μ
mu
gptkbp:mode μ
mu
gptkbp:moment_generating_function exp(mu*t + 0.5*sigma^2*t^2)
gptkbp:momentGeneratingFunction exp(μt + 0.5σ²t²)
gptkbp:moments all moments exist
gptkbp:namedAfter gptkb:Carl_Friedrich_Gauss
Carl_Friedrich_Gauss
gptkbp:parameter mean
standard deviation
variance
mu
sigma
sigma_squared
standard_deviation
gptkbp:pdf (1/(σ√(2π))) * exp(-0.5*((x-μ)/σ)^2)
gptkbp:probability_density_function f(x) = (1/(sigma*sqrt(2*pi))) * exp(-(x-mu)^2/(2*sigma^2))
gptkbp:relatedTo bell_curve
central_limit_theorem
chi_squared_distribution
error_function
log_normal_distribution
student_t_distribution
z_score
gptkbp:second_moment mu^2 + sigma^2
gptkbp:shape bell curve
gptkbp:skewness 0
gptkbp:special_case standard_normal_distribution
gptkbp:standard_deviation sigma
gptkbp:standard_normal_cdf Phi(x)
gptkbp:standard_normal_mean 0
gptkbp:standard_normal_pdf f(x) = (1/sqrt(2*pi)) * exp(-x^2/2)
gptkbp:standard_normal_standard_deviation 1
gptkbp:standard_normal_variance 1
gptkbp:standardDeviation σ
gptkbp:standardNormalCdf Φ(x)
gptkbp:standardNormalMean 0
gptkbp:standardNormalPdf (1/√(2π)) * exp(-0.5x²)
gptkbp:standardNormalStdDev 1
gptkbp:standardNormalVariance 1
gptkbp:subclassOf gptkb:standard_normal_distribution
gptkbp:supports (-∞, ∞)
x in (-infinity, infinity)
gptkbp:symmetry true
gptkbp:third_moment mu^3 + 3*mu*sigma^2
gptkbp:usedIn gptkb:machine_learning
gptkb:natural_sciences
gptkb:probability_theory
finance
social sciences
statistics
machine_learning
natural_sciences
gptkbp:variant σ²
sigma_squared
gptkbp:bfsParent gptkb:Wiener_process
gptkb:Boost.Random
gptkbp:bfsLayer 3
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