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In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function that describes the relative likelihood for this random variable to take on a given value. The probability of the random variable falling within a particular range of values is given by the integral of this variable’s density over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one. The terms "''probability distribution function''"〔(Probability distribution function ) PlanetMath〕 and "''probability function''"〔(Probability Function ) at Mathworld 〕 have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values, or it may refer to the cumulative distribution function, or it may be a probability mass function rather than the density. Further confusion of terminology exists because ''density function'' has also been used for what is here called the "probability mass function".〔Ord, J.K. (1972) ''Families of Frequency Distributions'', Griffin. ISBN 0-85264-137-0 (for example, Table 5.1 and Example 5.4)〕 ==Example== Suppose a species of bacteria typically lives 4 to 6 hours. What is the probability that a bacterium lives ''exactly'' 5 hours? The answer is actually 0%. A lot of bacteria live for ''approximately'' 5 hours, but there is a negligible chance that any given bacterium dies at ''exactly'' 5.0000000000... hours. Instead we might ask: What is the probability that the bacterium dies between 5 hours and 5.01 hours? Let's say the answer is 0.02 (i.e., 2%). Next: What is the probability that the bacterium dies between 5 hours and 5.001 hours? The answer is probably around 0.002, since this is 1/10th of the previous interval. The probability that the bacterium dies between 5 hours and 5.0001 hours is probably about 0.0002, and so on. In these three examples, the ratio (probability of dying during an interval) / (duration of the interval) is approximately constant, and equal to 2 per hour (or 2 hour−1). For example, there is 0.02 probability of dying in the 0.01-hour interval between 5 and 5.01 hours, and (0.02 probability / 0.01 hours) = 2 hour−1. This quantity 2 hour−1 is called the ''probability density'' for dying at around 5 hours. Therefore, in response to the question "What is the probability that the bacterium dies at 5 hours?", a literally correct but unhelpful answer is "0", but a better answer can be written as (2 hour−1) ''dt''. This is the probability that the bacterium dies within a small (infinitesimal) window of time around 5 hours, where ''dt'' is the duration of this window. For example, the probability that it lives longer than 5 hours, but shorter than (5 hours + 1 nanosecond), is (2 hour−1)×(1 nanosecond) ≃ 6×10−13 (using the unit conversion 3.6×1012 nanoseconds = 1 hour). There is a ''probability density function'' ''f'' with ''f''(5 hours) = 2 hour−1. The integral of ''f'' over any window of time (not only infinitesimal windows but also large windows) is the probability that the bacterium dies in that window. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「probability density function」の詳細全文を読む スポンサード リンク
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