Probability Distributions and Random Variables

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 

Obviously, Afo = 1 and My is the average value of the corresponding random variable. In what follows we will focus on the continuous case. The second moment is usually expressed by the variance. 

Following are some examples of frequently encountered probability distributions Poisson distribution. This is the discrete distribution 

Binomial distribution. This is a discrete distribution in finite space The probability that the random variable n takes any integer value between 0 and N is given by 

The normalization condition is satisfied by the binomial theorem since PW = (p + We discuss properties of this distribution in Section 7.3.3. 

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After defining fundamental terms used in probability and introducing set notation for events, we consider probability theorems facilitating tlie calculation of the probabilities of complex events. Conditional probability and tlie concept of independence lead to Bayes theorem and tlie means it provides for revision of probabilities on tlie basis of additional evidence. Random variables, llicir probability distributions, and expected values provide tlie means... 

Consider, in general, the overall problem consisting of m balances and divide it into m smaller subproblems, that is, we will be processing one equation at a time. Then, after the i th balance has been processed, a new value of the least squares objective (test function) can be computed. Let J, denote the value of the objective evaluated after the i th equation has been considered. The approach for the detection of a gross error in this balance is based on the fact that fa is a random variable whose probability distribution can be calculated. 

I> = 8c0ifi, and where I is the time spent at site i. When a random variable is defined as the sum of several independent random variables, its probability distribution is the convolution product of the distributions of the terms of the... 

For discrete random variables, the probability distribution can often be determined using mathematical intuition, as all experiments are characterized by a hxed set of outcomes. For example, consider an experiment in which a six-sided die is thrown. The variable x denotes the number on the die face, and P(x) is the probability distribution, i.e., the chance of observing x on the face following a throw. Since this random variable is discrete, if the die is fair, then the probability of any possible value is equal and is given by 1/n, where n is the number of sides, since the sum of all possible outcomes must be unity. The distribution is, therefore, called uniform. Hence, for n = 6, P(x) = 1/6, where x = 1, 2, 3, 4, 5, or 6. 

Discrete and continuous variables and probability distributions From Clause 5.3.3 of Chapter I, we get the probability mass function and cumulative distribution functions. For a single dimension, discrete random variable X, the discrete probability function is defined by/(xi), such that/(xi) > 0, for all xie R (range space), and f xi) = F(x) where F(x) is known as cumulative... 

Transformation n A functional replacement or change in variables of one or more variables. Transformations are often used to simplify the behavior or shape of a random variable s probability distribution, to create a constant variable, or simphfy the relationship between variables. Transformations are generally named by either their purpose or by their functional transformation. Two common transformations named for their purpose are the transformation to normality, whose purpose is to change the shape of a probability distribution so that it is closer to that of the normal distribution, and the transformation to linearity whose purpose is to change the relation between two variables so that it is linear. Common types of transformations named for their functional transformation are the logarithmic transformation, which replaces a variable by its logarithm, the 1/x transformation, which replaces a variable by its inverse, and the transformation, which replaces a variable by its square. 

Fig. 10.19 The probability density of the extreme value distribution typical of the MSP scores for random sequena The probability that a random variable with this distribution has a score of at least x is given by 1 - exp[-e -where u is the characteristic value and A is the decay constant. The figure shows the probability density function (which corresponds to the function s first derivative) for u = 0 and A = 1.

Property 1 indicates tliat tlie pdf of a discrete random variable generates probability by substitution. Properties 2 and 3 restrict the values of f(x) to nonnegative real niunbers whose sum is 1. An example of a discrete probability distribution function (approaching a normal distribution - to be discussed in tlie next chapter) is provided in Figure 19.8.1. 

Consider the following inventory problem. There are p time periods at the start of each of which an order of n items is made at a cost A(n), which is an increasing function of % (e.g., A(n) = a + bn). The length of each period is a random variable, and, hence, there are p random variables Xt (i = 1,- -, p) that are assumed to be independently and identically distributed according to the distribution function Fn(x)—for each period, it is the probability that there is a demand for... 

FIG U RE 6.3 Two parametric classes of prior distributions having constant variance (left) or constant mean (right) shown as cumulative distribution functions (cdfs). The horizontal axis is some value for a random variable and the vertical axis is (cumulative) probability. 

Normal Random Variable. The probability density function of a normally distributed random variable, y, is completely characterized by its arithmetic mean, y, and its standard deviation, a. This is abbreviated as N (y,cr2) and written as ... 

Lognormal Random Variable. Every normally distributed random variable, y, is uniquely associated with a lognormally distributed random variable, x, whose probability density function is completely characterized by its geometric mean, GM, and geometric standard deviation, GSD (2). 

The use of Monte Carlo and other stochastic analytical methods to characterize the distribution of exposure and dose-response relationships is increasing (IPCS, 2001a). The Monte Carlo method uses random numbers and probability in a computer simulation to predict the outcome of exposure. These methods can be important tools in risk characterization to assess the relative contribution of uncertainty and variability to a risk estimate. 

The goal of classification, also known as discriminant analysis or supervised learning, is to obtain rules that describe the separation between known groups of observations. Moreover, it allows the classification of new observations into one of the groups. We denote the number of groups by Z and assume that we can describe our experiment in each population icj by a / -dimensional random variable Xj with distribution function (density) fj. We write pj for the membership probability, i.e., the probability for an observation to come from icj. 

Tlie probability distribution of a random variable concerns die distribution of probability over die range of die random variable. The distribution of probability is specified by the pdf (probability distribution function). This section is devoted to general properties of die pdf in die case of discrete and continuous nmdoiii variables. Specitil pdfs finding e. tensive application in liazard and risk mialysis are considered in Cliapter 20. 

Where /and h are known nonlinear functions. The random variables and Vh represent the process and measurement noises, respectively. They are assumed to be independent, white noises with normal probability distributions p(w) N(0,Q) andp(v N(0Jl). 

In Chapter 5 we described a number of ways to examine the relative frequency distribution of a random variable (for example, age). An important step in preparation for subsequent discussions is to extend the idea of relative frequency to probability distributions. A probability distribution is a mathematical expression or graphical representation that defines the probability with which all possible values of a random variable will occur. There are many probability distribution functions for both discrete random variables and continuous random variables. Discrete random variables are random variables for which the possible values have "gaps." A random variable that represents a count (for example, number of participants with a particular eye color) is considered discrete because the possible values are 0, 1, 2, 3, etc. A continuous random variable does not have gaps in the possible values. Whether the random variable is discrete or continuous, all probability distribution functions have these characteristics ... 

A simple example of a discrete probability distribution is the process by which a single participant is assigned the active treatment when the event "active treatment" is equally likely as the event "placebo treatment." This random process is like a coin toss with a perfectly fair coin. If the random variable, X, takes the value of 1 if active treatment is randomly assigned and 0 if the placebo treatment is randomly assigned, the probability distribution function can be described as follows ... 

Every possible outcome of a random variable is associated with a probability for that event occurring. Two functions map outcome to probability for continuous random variables the probability density function (pdf) and cumulative distribution function (cdf). In the discrete case, the pdf and cdf are referred to as the probability mass function and cumulative mass function, respectively. A function f(x) is a pdf for some continuous random variable X if and only if... 

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