Chikvadratsfordelingen

Chikvadratsfordelingen, often shortened to chikvadratfordeling, is a fundamental probability distribution in statistics. It arises from the sum of squared independent standard normal random variables. If Z1, Z2, ..., Zk are independent standard normal random variables, then their sum of squares, X = Z1^2 + Z2^2 + ... + Zk^2, follows a chikvadratsfordeling with k degrees of freedom. The parameter k, known as the degrees of freedom, determines the shape of the distribution.

The chikvadratsfordeling is always non-negative, meaning its values range from zero to infinity. It is skewed to the right, with the skewness decreasing as the degrees of freedom increase. For a small number of degrees of freedom, the distribution is highly skewed. As the degrees of freedom become large, the chikvadratsfordeling approximates a normal distribution.

The probability density function (PDF) of a chikvadratsfordeling with k degrees of freedom is given by f(x; k) = (1 / (2^(k/2) * Gamma(k/2))) * x^((k/2)-1) * e^(-x/2) for x > 0, and f(x; k) = 0 for x <= 0. Here, Gamma(k/2) is the gamma function.

The chikvadratsfordeling is widely used in hypothesis testing, particularly in tests of independence and goodness-of-fit. The chi-squared test is a prime example of its application, where it is used to determine if there is a statistically significant association between categorical variables or if an observed distribution matches an expected distribution.