Gaussian distribution

A continuous probability distribution with a bell-shaped density determined by its mean and variance. Its values are symmetric around the mean, with probabilities concentrated near it.

Probability density function: A function whose values describe the relative likelihood of outcomes in a continuous probability distribution. The Gaussian density assigns probability across the real line, though probability at any single point is zero.
Normal probability density function: The formula giving the density of a normal distribution at each real-valued point. Its exponential form makes the mean and variance determine the entire curve.
Central limit theorem: A theorem stating that suitably standardized sums of many independent variables often approach a normal distribution. It explains why Gaussian approximations arise even when individual measurements are not Gaussian.
Student's t-distribution: A symmetric continuous distribution with heavier tails than the standard normal distribution. Its heavier tails better accommodate uncertainty from estimating variance with limited data.
Expected value: The probability-weighted average value of a random variable. For a Gaussian distribution, the expected value is its center and symmetry point.
Cumulative distribution function: A function giving the probability that a random variable is at most a specified value. It converts the Gaussian density into probabilities and quantiles.
Linear regression: A method for modeling a response as a linear function of predictors and random error. Gaussian error assumptions yield familiar estimation and uncertainty formulas.
Laplace distribution: A continuous distribution with a symmetric, sharply peaked density that decays exponentially. Its sharper center and heavier tails offer a different model of symmetric noise.
Variance: The expected squared deviation of a random variable from its mean. Variance determines the Gaussian distribution’s spread around its center.
Quantile: A value below which a specified proportion of a probability distribution lies. Gaussian quantiles locate thresholds such as central confidence intervals.

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