Continuous random variables, probability

This continuous random variable probability distribution is named after its founder, John Rayleigh (1842-1919), and its probability density function is defined by [1,8]... 

This continuous random variable probability distribution was developed in the early 1950s by Walliodi Weibull, a Swedish professor in mechanical engineering [15]. The probability density function for the distribution is defined by... 

Property 1 indicates diat die pdf of a continuous random variable generates probability by integradon of the pdf over die interval whose probability is required. Wlien diis interval contracts to a single value, die integral over the... 

Anotlier fimction used to describe tlie probability distribution of a random variable X is tlie cumulative distribution function (cdf). If f(x) specifies tlie pdf of a random variable X, tlien F(x) is used to specify the cdf For both discrete and continuous random variables, tlie cdf of X is defined by ... 

Moments 92. Common Probability Distributions for Continuous Random Variables 94. Probability Distributions for Discrete Random Variables. Univariate Analysis 102. Confidence Intervals 103. Correlation 105. Regression 106. 

Common Probability Distributions for Continuous Random Variables... 

A bounded continuous random variable with uniform distribution has the probability function... 

The chi-square distribution gives the probability for a continuous random variable bounded on the left tail. The probability function has a shape parameter... 

The density function of the sum of two independent continuous random variables is computed by the convolution of the two probability densities. Loosely speaking, two random numbers are independent, if they do not influence each other. Unfortunately, convolutions are obviously important but not convenient to calculate. 

In most natural situations, physical and chemical parameters are not defined by a unique deterministic value. Due to our limited comprehension of the natural processes and imperfect analytical procedures (notwithstanding the interaction of the measurement itself with the process investigated), measurements of concentrations, isotopic ratios and other geochemical parameters must be considered as samples taken from an infinite reservoir or population of attainable values. Defining random variables in a rigorous way would require a rather lengthy development of probability spaces and the measure theory which is beyond the scope of this book. For that purpose, the reader is referred to any of the many excellent standard textbooks on probability and statistics (e.g., Hamilton, 1964 Hoel et al., 1971 Lloyd, 1980 Papoulis, 1984 Dudewicz and Mishra, 1988). For most practical purposes, the statistical analysis of geochemical parameters will be restricted to the field of continuous random variables. 

Figure 4.1 Relationship between the probability density function f x) of the continuous random variable X and the cumulative distribution function F(x). The shaded area under the curve f(x) up to x0 is equal to the value of f x) at x0. 

Probability density function (PDF) The PDF is referred to as the probability function or the frequency function. For continuous random variables, that is, the random variables that can assume any value within some defined range (either finite or infinite), the probability density function expresses the probability that the random variable falls within some very small interval. For... 

The fundamental probability transformation. Suppose that the continuous random variable x has cumulative distribution F(x). What is the probability distribution of the random variable y = F(x)1 (Observation This result forms the basis of the simulation of draws from many continuous distributions.)... 

The chi distribution is often confused with and used to describe the chi-square distribution or x2 distribution which is the distribution of the continuous random variable that represents the sum of the normalized squares of the X. random variables. This is equal to the probability distribution that describes the square of the chi distribution, Y2, which is given by ... 

A third measure of location is the mode, which is defined as that value of the measured variable for which there are the most observations. Mode is the most probable value of a discrete random variable, while for a continual random variable it is the random variable value where the probability density function reaches its maximum. Practically speaking, it is the value of the measured response, i.e. the property that is the most frequent in the sample. The mean is the most widely used, particularly in statistical analysis. The median is occasionally more appropriate than the mean as a measure of location. The mode is rarely used. For symmetrical distributions, such as the Normal distribution, the mentioned values are identical. 

Relative likelihood indicates the chance that a value or an event will occur. If the random variable is a discrete random variable, then the relative likelihood of a value is the probability that the random variable equals that value. If the random variable is a continuous random variable, then the relative likelihood at a value is the same as the probability density function at that value. 

On the other hand, random errors do not show any regular dependence on experimental conditions, since they are generated by many small and uncontrolled causes acting at the same time, and can be reduced but not completely eliminated. Thus, random errors are observed when the same measurement is repeatedly performed. In the simplest case, the universe of random errors is described by a continuous random variable e following a normal distribution with zero mean, i.e., for a univariate variable, the probability density function is given by... 

A probability distribution is a mathematical description of a function that relates probabilities with specified intervals of a continuous quantity, or values of a discrete quantity, for a random variable. Probability distribution models can be non-parametric or parametric. A non-parametric probability distribution can be described by rank ordering continuous values and estimating the empirical cumulative probability associated with each. Parametric probability distribution models can be fit to data sets by estimating their parameter values based upon the data. The adequacy of the parametric probability distribution models as descriptors of the data can be evaluated using goodness-of-fit techniques. Distributions such as normal, lognormal and others are examples of parametric probability distribution models. 

A function that relates probability density to point values of a continuous random variability or that relates probability to specific categories of a discrete random variable. The integral (or sum) must equal one for continuous (discrete) random variables. 

A probability distribution function for a continuous random variable, denoted by fix), describes how the frequency of repeated measurements is distributed over the range of observed values for the measurement. When considering the probability distribution of a continuous random variable, we can imagine that a set of such measurements will lie within a specific interval. The area under the curve of a graph of a probability distribution for a selected interval gives the probability that a measurement will take on a value in that interval. 

For scalar continuous random variables X and Y with joint probability density f (x, y), marginals and conditionals are refined as... 

Probability density function (pdf) Indicates the relative likelihood of the different possible values of a random variable. For a discrete random variable, say X, the pdf is a function, say /, such that for any value x, /(x) is the probability that X = X. For example, if X is the number of pesticide applications in a year, then /(2) is the probability density function at 2 and equals the probability that there are two pesticide applications in a year. For a continuous random variable, say Y, the pdf is a function, say g, such that for any value y, g(y) is the relative likelihood that Y = y,0 < g y), and the integral of g over the range of y from minus infinity to plus infinity equals 1. For example, if Y is body weight, then g(70) is the probability density function for a body weight of 70 and the relative likelihood that the body weight is 70. Furthermore, if g 70)/g(60) = 2, then the body weight is twice as likely to be 70 as it is to be 60 (Sielken, Ch. 8). 

The distribution of a population s property can be introduced mathematically by the repartition function of a random variable. It is well known that the repartition function of a random variable X gives the probability of a property or event when it is smaller than or equal to the current value x. Indeed, the function that characterizes the density of probability of a random variable (X) gives current values between X and x -I- dx. This function is, in fact, the derivative of the repartition function (as indirectly shown here above by relation (5.16)). It is important to make sure that, for the characterization of a continuous random variable, the distribution function meets all the requirements. Among the numerous existing distribution functions, the normal distribution (N), the chi distribution (y ), the Student distribution (t) and the Fischer distribution are the most frequently used for statistical calculations. These different functions will be explained in the paragraphs below. 

The error of an observation in the nth event of a sequence may be modeled as a random variable e . For a continuous random variable with range —oo.oo), a commonly used probability density model is the normal error curve... 

Probability Density Continuous Random Variables and Probability Functions... 

The probability distribution of a random variable concerns the distribution of probability over the range of the random variable. The distribution of probability is specified by the probability distribution function (PDF). The random variable may be discrete or continuous. Special PDFs finding application in risk analysis are considered in later problems. The PDF of a continuous random variable X has the following properties ... 

A random variable is an observable whose repeated determination yields a series of numerical values ( realizations of the random variable) that vary from trial to trial in a way characteristic of the observable. The outcomes of tossing a coin or throwing a die are familiar examples of discrete random variables. The position of a dust particle in air and the lifetime of a light bulb are continuous random variables. Discrete random variables are characterized by probability distributions P denotes the probability that a realization of the given random variable is n. Continuous random variables are associated with probability density functions P(x) P(xi)dr... 

---