Discrete probability distributions random

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. 

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 ... 

Since distributions describing a discrete random variable may be less familiar than those routinely used for describing a continuous random variable, a presentation of basic theory is warranted. Count data, expressed as the number of occurrences during a specified time interval, often can be characterized by a discrete probability distribution known as the Poisson distribution, named after Simeon-Denis Poisson who first published it in 1838. For a Poisson-distributed random variable, Y, with mean X, the probability of exactly y events, for y = 0,1, 2,..., is given by Eq. (27.1). Representative Poisson distributions are presented for A = 1, 3, and 9 in Figure 27.3. 

In this section we describe the six discrete probability distributions and five continuous probability distributions that occur most frequently in bioinformatics and computational biology. These are called univariate models. In the last three sections, we discuss probability models that involve more than one random variable called multivariate models. 

We consider a minimization rather than a maximization problem for the sake of notational convenience.) Here C R is a set of permissible values of the vector x of decision variables and is referred to as the feasible set of problem (11). Often x is defined by a (finite) number of smooth (or even linear) constraints. In some other situations the set x is finite. In that case problem (11) is called a discrete stochastic optimization problem (this should not be confused with the case of discrete probability distributions). Variable random vector, or in more involved cases as a random process. In the abstract fiamework we can view as an element of the probability space (fi, 5, P) with the known probability measure (distribution) P. 

Binomial Distribution A distribution of data or results describing probabilities of the outcome of trials that can have one or two mutually exclusive results (e.g., exposure above or below a permissible exposure limit or PEL ). This theoretically discrete probability distribution for a binomial random variable is represented as ... 

Note that X is called a random variable. The function (X) is a discrete probability distribution function. By definition... 

The probability distribution of a randoni variable concerns tlie distribution of probability over tlie range of tlie random variable. The distribution of probability is specified by the pdf (probability distribution function). This section is devoted to general properties of tlie pdf in tlie case of discrete and continuous nmdoiii variables. Special pdfs finding e.xtensive application in liazard and risk analysis are considered in Chapter 20. 

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. 

An order density is a demand density 5 with 5(0) = 0. The number of orders per interval can be described by a discrete density function t] with discrete probabilities defined for nonnegative integers 0,1, 2, 3, —The resulting (t], 8)-compounddensity Junction is constructed as follows A random number of random orders constitute the random demand. The random number of orders is r -distributed. The random orders are independent from the number of orders, and are independent and identically 5-distributed. 

A stochastic program is a mathematical program (optimization model) in which some of the problem data is uncertain. More precisely, it is assumed that the uncertain data can be described by a random variable (probability distribution) with sufficient accuracy. Here, it is further assumed that the random variable has a countable number of realizations that is modeled by a discrete set of scenarios co = 1,..., 2. 

There are two different ways of representing uncertainty. The first approach is the continuous probability distribution where numerical integration is employed over the random continuous probability space. This approach maintains the model size but on the other hand introduces nonlinearities and computational difficulties to the problem. The other approach is the scenario-based approach where the random space is considered as discrete events. The main disadvantage of this approach is the substantial increase in computational requirements with an increase in the number ofuncertain parameters. The discrete distribution with a finite number K of possible... 

The simplest of these models which permits a detailed discussion of the decay of correlations is a random walk model in which a set of random walkers whose positions are initially correlated is allowed to diffuse the motion of any single random walker being independent of any other member of the set. Let us assume that there are r particles in the set and motion occurs on a discrete lattice. The state of the system is, therefore, completely specified by the probabilities Pr(nlf n2,..., nr /), (tij = — 1, 0, 1, 2,. ..) in which Pr(n t) is the joint probability that particle 1 is at n1( particle 2 is at n2, etc., at time l. We will also use the notation Nj(t) for the random variable that is the position of random walker j at time t. Reduced probability distributions can be defined in terms of the Pr(n t) by summation. We will use the notation P nh, rth,..., ntj I) to denote the distribution of random walkers iu i2,..., i at time t. We define... 

A single coin is an example of a Bernoulli" distribution. This probability distribution limits values of the random variable to exactly two discrete values, one with probability p, and the other with the probability (1-p). For the coin, the two values are heads p, and tails (1-p), where p = 0.5 for a fair coin. 

A discrete distribution function assigns probabilities to several separate outcomes of an experiment. By this law, the total probability equal to number one is distributed to individual random variable values. A random variable is fully defined when its probability distribution is given. The probability distribution of a discrete random variable shows probabilities of obtaining discrete-interrupted random variable values. It is a step function where the probability changes only at discrete values of the random variable. The Bernoulli distribution assigns probability to two discrete outcomes (heads or tails on or off 1 or 0, etc.). Hence it is a discrete distribution. 

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 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... 

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 ... 

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