Therefore, X is a discrete random variable if as u runs through all possible values of the random variable X. A random variable is discrete if its probability distribution is discrete and can be characterized by a PMF.
The PMF differs from the PDF in that the values of the latter, defined only for continuous random variables, are not probabilities rather, its integral over a set of possible values of the random variable is a probability. In probability theory, a probability mass function, or PMF, gives the probability that a discrete random variable is exactly equal to some value. Every CDF is monotonically increasing, is continuous from the right, and at the limits has the following properties:įurther, the CDF is related to the PDF by, where the PDF function f is the derivative of the CDF function F. The cumulative distribution function (CDF) is denoted as F(x) = P(X ≤ x) indicating the probability of X taking on a less than or equal value to x. The probability, in reality, is the function f(x)dx discussed previously, where dx is an infinitesimal amount. The random variable x within this distribution will have f(x) greater than 1.
In fact, f(a) can sometimes be larger than 1––consider a uniform distribution between 0.0 and 0.5. It is a common mistake to think of f(a) as the probability of a. The probability of the interval between is given by, which means that the total integral of the function f must be 1.0. The PDF, therefore, can be seen as a smoothed version of a probability histogram that is, by providing an empirically large sample of a continuous random variable repeatedly, the histogram using very narrow ranges will resemble the random variable’s PDF. If a probability distribution has a density of f(x), then intuitively the infinitesimal interval of has a probability of f(x) dx. In mathematics and Monte Carlo risk simulation, a probability density function (PDF) represents a continuous probability distribution in terms of integrals. Finally, the inverse cumulative distribution function (ICDF) is used to compute the value x given the cumulative probability of occurrence. In addition, the cumulative distribution function (CDF) can also be computed, which is the sum of the PDF values up to this x value. This paper briefly explains the probability density function (PDF) for continuous distributions, which is also called the probability mass function (PMF) for discrete distributions (we use these terms interchangeably), where given some distribution and its parameters, we can determine the probability of occurrence given some outcome or random variable x. Comments Off on Tips on Interpreting PDF, CDF, and ICDF.