Hajufunktsioon

Hajufunktsioon, also known as a distribution function or cumulative distribution function (CDF), is a fundamental concept in probability theory and statistics. It is used to describe the probability distribution of a random variable. For a given random variable X, the hajufunktsioon F(x) is defined as the probability that X is less than or equal to a specified value x. Mathematically, it is expressed as:

F(x) = P(X ≤ x)

The hajufunktsioon has several key properties. It is non-decreasing, meaning that as x increases, F(x) does not decrease. It is also right-continuous, meaning that the limit of F(x) as x approaches a value from the right is equal to F(x) itself. Additionally, the hajufunktsioon approaches 0 as x approaches negative infinity and 1 as x approaches positive infinity.

The hajufunktsioon is particularly useful in statistics for summarizing the distribution of a dataset. It can be used to calculate various statistical measures, such as the median and quartiles, and to compare different distributions. It is also a crucial component in the theory of probability, where it is used to define the probability of events and to derive other probability distributions.

In practice, the hajufunktsioon is often estimated from sample data using the empirical distribution function. This is a step function that jumps up by 1/n at each of the n observed data points, where n is the sample size. The empirical distribution function is a non-parametric estimate of the true hajufunktsioon and is used to make inferences about the underlying population distribution.