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In probability theory and statistics, the terms DF (distribution function), CDF (cumulative distribution function), and PF (probability function) refer to different ways of describing the distribution of random variables.
1. PDF (Probability Density Function): This applies to continuous random variables and describes the likelihood of the variable taking on a specific value. The area under the PDF curve over a specific interval gives the probability of the random variable falling within that interval.
2. CDF (Cumulative Distribution Function): This function gives the probability that a random variable takes on a value less than or equal to a specific value. It is the integral of the PDF for continuous variables or the sum of the probabilities (from the PF) for discrete variables. The CDF is always non-decreasing and ranges from 0 to 1.
3. DF (Distribution Function): This term is often used interchangeably with CDF but can also refer to any distribution-related function describing characteristics of a random variable. However, it’s more common to see the term CDF used specifically to denote the cumulative distribution function.
In summary, the relationship among these functions is as follows:
– The PF gives the probabilities for each discrete outcome.
– The CDF is derived from the PF by summing the probabilities up to a value for discrete variables, or integrating the PDF for continuous variables.
– The PDF represents probabilities for continuous variables and can be integrated to yield the CDF.
Ultimately, these functions are interconnected and help