Basic concepts of probability and statistics

Basic terms and definition

  • Random variable X

A variable that takes values with certain probability, described by a probability distribution.

  • Observation x

An observed value of the variable X.

  • Sample

A set of observations x1, x2, ..., xn.

  • Population

A finite or infinite set from which samples are drawn. Population has constant statistical properties while properties of samples may vary from one to another.

  • Sample space

The set of all samples that could be drawn from the population, usually denoted W. Sample space is in fact the domain of definition of the variable X.

  • Random event A

A subset of the sample space.

Random variables are:

  • discrete (sample space = set of integer numbers), or
  • continuous (sample space = set of real numbers).

Examples of discrete hydrologic random variables are number of days within a year having daily precipitation higher than 10 mm, number of days within a year having water stages above 300 cm, or number of floods with peak flows exceeding certain threshold. Continuous hydrologic random variables are flow rate, water stage or level, groundwater level, rainfall depth or volume, etc.

Probability of an event, P{A}, is the chance that the event A will occur when the observation is made. Probabilities of events can be estimated from a sample with total of n observations and with nA values in the range of event A. Then the relative frequency of A is nA/n. With increasing sample size n, relative frequency is better estimate of the event probability P{A}, that is:


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Hydrology of Floods






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