Normalijakautuma

Normalijakautuma, known in Finnish as normaalijakauma, is a continuous probability distribution that is symmetric and bell-shaped. It is fully described by two parameters: the mean μ, which determines the center, and the standard deviation σ > 0, which determines the spread. The probability density function is f(x) = (1/(σ√(2π))) exp(-(x-μ)²/(2σ²)). The cumulative distribution function is F(x) = (1/2)[1 + erf((x-μ)/(σ√2))], where erf denotes the error function.

Key properties include symmetry around μ, zero skewness, and a kurtosis of 3 (excess kurtosis 0). The mean is μ and the variance is σ². Linear transformations preserve normality: if X ~ N(μ, σ²), then aX + b ~ N(aμ + b, a²σ²). The distribution is central to the Central Limit Theorem, which explains why sums of many independent variables with finite mean and variance tend to be normally distributed, making the normal distribution a common approximation in statistics. A standard normal variable Z has μ = 0 and σ = 1; any normal variable can be standardized by Z = (X-μ)/σ.

Applications include modeling measurement errors, natural phenomena, and various statistical methods that rely on normality assumptions, such as confidence intervals and hypothesis tests. Estimation of μ and σ is typically done via the sample mean and sample standard deviation, and maximum likelihood estimation yields the same estimators under normality.

Limitations include that real data may deviate from normality, especially for bounded, skewed, or heavy-tailed variables. In such cases, alternative distributions or data transformations may be more appropriate.