Probability distribution discrete

Figure 2.5-1 illustrates the fact that probabilities are not precisely known but may be represented by a "bell-like" distribution the amplitude of which expresses the degree of belief. The probability that a system will fail is calculated by combining component probabilities as unions (addition) and intersection (multiplication) according to the system logic. Instead of point values for these probabilities, distributions are used which results in a distributed probabilitv of system fadure. This section discusses several methods for combining distributions, namely 1) con olution, 2i moments method, 3) Taylor s series, 4) Monte Carlo, and 5) discrete probability distributions (DPD). 

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. 

This case has been dealt with for continuous (rather than discrete) probability distributions in an earlier paper.27 Also, it is worth noting that the physics of this process are similar to the process of coagulation in atmospheric physics.37... 

For a discrete probability distribution, we can define a function f which gives the probability P of getting the outcome A=x, where A is the discontinuous variable. 

A.4.11 Discrete probability distributions model systems with finite, or countably infinite, values, while a continuous probability distribution model systems with infinite possible values within a range. 

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. 

This example demonstrates the simple method by which a discrete probability distribution can be determined. 

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. 

Discrete probability distributions include the Binomial distribution and the Poisson distribution. 

Once the BN has been built (nodes, arcs and relative CPTs), it can be used by the analysts to assess the dependence level. The model requires as input from the analyst a discrete probability distribution P(X = X ) for each k-th input factor, k = 1,2,..., m with Xf defining the actual state of the k-th input factor and X, V = 1,2,..., the possible input states. These input probability distributions are combined with the CPTs in order to compute a discrete probability distribution of the output factor y P(Y = 7 ), v =... 

For each input factor x, the corresponding anchors are characterized in terms of a discrete probability distribution over the states X. Figures 3 and 4 show the... 

The probability values assigned by this approach are then used directly as discrete probability distributions for the input factors of the BN. 

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