It also allows, through a sum (an integral) to give the probability of a series of events (described by an interval) to occur. While the measure does not mean much by itself, it allows to compare the relative chances to occur of two events. In the case of continuous variables, each (possible) event has a non-zero probability of occurring, but it is so small that only the probability distribution makes it possible to measure it. In the case of discrete variables, each event has a non-zero probability as long as it is possible. In this case, we will match the event "there are 10 people queuing" with the number 10, or "the temperature is 59.8☏" with the real 59.8.
While this is perfectly arbitrary in the case of a coin, it may be more natural if one counts the number of people in a queue at the post office, or if one measures the air temperature. For example, we can match the tossing of a coin with 0 if the coin lands on head, and 1 if the coin lands on tail.
Nevertheless, it is so complex that it is preferable to estimate that each face has a certain probability of being the result of the throw and that the occurrence of an event is random.Ī random variable is also a function that maps an event to a real. When you roll a die, a perfect knowledge of the starting movement allow you knowing when, where, and in what position the die would stop. Randomness is a representation of ignorance or imperfect knowledge. It allows to relate events (for example, the occurrence of a 2 when throwing a die), to their probability.Ī large number of probability distributions have been developed to describe particular situations where randomness occurs. A probability distribution is a special type of function, that is named a measure in mathematics.
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