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Statistics for Quality
Published in K. S. Krishnamoorthi, V. Ram Krishnamoorthi, Arunkumar Pennathur, A First Course in Quality Engineering, 2018
K. S. Krishnamoorthi, V. Ram Krishnamoorthi, Arunkumar Pennathur
Although a random variable may seem to behave in a haphazard or chaotic manner to an ordinary observer, statisticians have found that they follow certain patterns and that the patterns can be captured in mathematical models. These models, represented by formulas, graphs, or tables, are called “probability distributions.” Because the two types of random variables mentioned above are mathematically different, one being continuous and the other non-continuous, they need two different types of models. The model used to describe a discrete random variable is called the “probability mass function,” and that used to describe a continuous random variable is called the “probability density function.” First, we will discuss the probability distribution of a discrete random variable: the probability mass function.
Probability, Random Variables, and Stochastic Processes
Published in Erchin Serpedin, Thomas Chen, Dinesh Rajan, Mathematical Foundations for SIGNAL PROCESSING, COMMUNICATIONS, AND NETWORKING, 2012
Random variable X is said to be a discrete random variable if the CDF is constant except at a countable set of points. For a discrete random variable, the probability mass function (PMF), PX (x), is equal to the probability that random variable X takes on value x. Thus PX (x) = FX (x) − FX (x−). Clearly, since the PMF represents a probability value, PX (x) ≥ 0 and ∑xPX(x)=1. Similar to the CDF, the PMF also completely determines the properties of a discrete random variable.
Visualizing Categorical Data
Published in Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka, Exploratory Data Analysis with MATLAB®, 2017
Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka
With discrete variables, we have a probability mass function, rather than a density function. The probability mass function assigns a probability to the event that the random variable will take on specific discrete values in the domain. Two of the most common discrete distributions are the binomial and the Poisson [Martinez and Martinez, 2015].
An expectation operator for belief functions in the Dempster–Shafer theory*
Published in International Journal of General Systems, 2020
In probability theory, for discrete real-valued random variables characterized by a probability mass function (PMF), the expected value of X can be regarded as a weighted average of the states of X where the weights are the probabilities associated with the values. Our definition is similar. As we have probabilities associated with subsets of states, first we define the value of a subset as the weighted average of the states of the subset where the weights are the commonality values of the singleton states. Then the expected value of X is defined to be the weighted average of the values of the subsets where the weights are the commonality values of the subsets.
Reliability assessment of ageing infrastructures: an interdisciplinary methodology
Published in Structure and Infrastructure Engineering, 2020
Andreas Panenka, François Marie Nyobeu Fangue, Rolf Rabe, Heike Schmidt-Bäumler, Julia Sorgatz
Processing quantitative data is necessary in order to mathematically describe the relevant variables of the observed real-world situation and to translate the data into a mathematical model. Variables derived from measurements are realised within a range of possible, but random values. Probability functions transfer these random values into a mathematical model. A discrete random variable is described by its probability mass function; a continuous random variable by its probability density function (Ang & Tang, 2007).
Modelling the spatial distribution of heavy vehicle loads on long-span bridges based on undirected graphical model
Published in Structure and Infrastructure Engineering, 2019
Zhicheng Chen, Yuequan Bao, Jiahui Chen, Hui Li
In probability and statistics, the probability distribution of discrete random variable is described by probability mass function (PMF). The joint PMF of the vehicle-location model is a multivariate function gives the probability of each possible combination of realizations for the random vector given in Equation (2).