Distribution classes

Most general classes of distributions are discrete and continuous. Discrete distribution is restricted to finite or countable number of points. Continuous distribution is spread on some real interval.

Distribution characteristics

Probability distribution may be described by several equivalent tools. Some of them are presented below.

• Cumulative distribution function or CDF. This characteristic is defined for any real distribution. Let be a random variable. Its cumulative distribution function is calculated as , .
• Density function. This function is defined only for continuous distributions; it is the derivative of CDF: , .
• Probability function (discrete density). This function is used for discrete distributions. If distribution of a random variable is restricted to finite or countable set of points , , ..., , ..., then it might be describes by probabilities for the points:

  Value			...		...
  Probability			...		...

  Here p_k = f(x_k) = P(X = x_k), k=1,2,...,n,...

• Expectation (mean value) of a random variable . This is integral of the form . For continuous random variable may be written using density function , and for discrete random variable - via probability function: .
• Variance of a random variable is equal to
  .

  In classic risk theory variance was often used as a risk measure.

• Standard deviation of a random variable is calculated as .
• Asymmetry of a random variable : . characterizes the difference between tails of a distribution. Asymmetry is positive when right tail is heavier, and is negative when left tail is heavier. Asymmetry of symmetric distributions is equal to 0 (if exists).
• Kurtosis of a random variable : . describes heaviness of tails. Positive kurtosis indicates more heavy tails than those of normal distribution.
• Median of a random variable is a root of equation . Median is a mean characteristic of a distribution in the sence that takes values to the left and to the right of a with equal probabilities. Sometimes expectation is undefined, as in case of Cauchy distribution. In this case median is a natural descriptor of mean value. Median of discrete distribution may be not uniquely defined. For symmetric distributions median is equal to expectation (if the latter exists).
• Mode of a random variable is its most probable value: a point of maximum of a density function for continuous distributions, and a point of maximum of probability function for discrete distributions. Mode may be ambigous, so its usage in risk theory is restricted.

• Characteristic function, which is defined for any distribution. , . Here i is imaginary unit, a square root of (-1). Characteristic function of a continuous distribution may be written via density function: , , while that of a discrete ditribution is representable via probability function , .