Continuous random variable

Not everything in life is discrete. Many examples occur that are not, for example 

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Suppose that X denotes the time to failure of a bus section in an electrostatic precipitator. Then X is a continuous random variable whose range consists of the real numbers greater tlian zero. 

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 ... 

Figure 19.8.5. Graph of tlie cdf of a continuous random variable X.

In die case of a continuous random variable, die cdf is a continuous function. Suppose, for e.xample, dial X is a continuous random variable with pdf specified by... 

Tliese properties apply to tlie cases of botli discrete and continuous random variables. 

To illustrate the computation of variance and its interpretation in tlie case of continuous random variables, consider a random variable X liaving pdf specified by ... 

In tlie case of a random sample of observations on a continuous random variable assumed to have a so-called nonnal pdf, tlie graph of which is a bellshaped curve, tlie following statements give a more precise interpretation of the sample standard deviation S as a measure of spread or dispersion. 

Recalling tliat tlie derivative of the cdf of a continuous random variable is equal to its pdf permits rewriting Eq. (20.3.11) as... 

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. 

A discrete random variable is one that may take on only distinct, usually integer, values. A continuous random variable is one that may take on any value within a continuum of values. 

The moments describe the characteristics of a sample or distribution function. The mean, which locates the average value on the measurement axis, is the first moment of values measured about the origin. The mean is denoted by p for the population and X for the sample and is given for a continuous random variable by... 

The skew, the third moment about the mean, is a measure of symmetry of distribution and can be denoted by y (population) or g (sample). It is given for a continuous random variable by... 

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. 

A function applied in statistics to predict the relative distribution if the frequency of occurrence of a continuous random variable (i.e., a quantity that may have a range of values which cannot be individually predicted with certainty but can be described probabilistically) from which the mean and variance can be estimated. 

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... 

A final point about factors. They need not be continuous random variables. A factor might be the detector used on a gas chromatograph, with values flame ionization or electron capture. The effect of changing the factor no longer has quite the same interpretation, but it can be optimized— in this case simply by choosing the best detector. 

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. 

A continuous random variable x has a normal distribution with certain parameters ji (mean, parameter of location) and a2 (variance, parameter of spread) if its density function is given by the following equation ... 

Polymer growth is an example of a hybrid stochastic process, which involves both discrete and continuous random variables the position of the polymer being continuous, while the number of monomers in the growing polymer is discrete. In this section, we will discuss just those processes. 

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 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. 

Consider the situation in which a chemist randomly samples a bin of pharmaceutical granules by taking n aliquots of equal convenient sizes. Chemical analysis is then performed on each aliquot to determine the concentration (percent by weight) of pseudoephedrine hydrochloride. In this example, measurement of concentration is referred to as a continuous random variable as opposed to a discrete random variable. Discrete random variables include counted or enumerated items like the roll of a pair of dice. In chemistry we are interested primarily in the measurement of continuous properties and limit our discussion to continuous 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. 

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