Discrete random variables probability distributions

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. 

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. 

Cumulative distribution function of a discrete random variable Probability mass function of a discrete random variable Derivative of Gx(z) with respect to z... 

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. 

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. 

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. 

The Poisson distribution can be used to determine probabilities for discrete random variables where the random variable is the number of times that an event occurs in a single trial (unit of lime, space, etc.). The probability function for a Poisson random variable is... 

Cumulative distribution function (CDF) The CDF is referred to as the distribution fnnction, cumulative frequency function, or the cnmnlative probability fnnction. The cumnlative distribution fnnction, F(x), expresses the probability that a random variable X assumes a value less than or eqnal to some valne x, F(x) = Prob (X > x). For continnons random variables, the cnmnlative distribution function is obtained from the probability density fnnction by integration, or by snmmation in the case of discrete random variables. 

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. 

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 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 ... 

The probability of each outcome of four random treatment assignments is displayed in Table 6.2. In some instances, we may be interested in knowing what the probability of observing x or fewer successes would be, that is, P(A < x). This cumulative probability is also displayed for each outcome in Table 6.2. For a discrete random variable distribution, the sum of probabilities of each outcome must sum to 1, or unity. 

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. 

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. 

Suppose that annual operating savings, A, is a discrete random variable with probabilities as given in Table 9. The associated cumulative distribution function (CDF), also given in the table, represents the probability that the annual operating savings will be less than or equal to some given vine. 

In financial mathematics random variables are used to describe the movement of asset prices, and assuming certain properties about the process followed by asset prices allows us to state what the expected outcome of events are. A random variable may be any value from a specified sample space. The specifica-ti(Mi of the probability distribution that appUes to the sample space will define the frequency of particular values taken by the random variable. The cumulative distribution function of a random variable X is defined using the distribution function yo such that Pr Xdiscrete random variable is one that can assume a finite or countable set of values, usually assumed to be the set of positive integers. We define a discrete random variable X with its own proba-bdity function p i) such that p i)=Pr X = /. In this case the probabiUty distribution is... 

Poisson distribution A probability distribution for a discrete random variable. It is defined, for a variable (r) that can take values in the range 0,1,2,. and has a mean value p, as... 

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... 

Probability distribution function, pdf, of a discrete random variable A... 

Geometric Distribution n A probability distribution of a discrete random variable, X, with non-negative integer values, k, that have a probability, P(/c), of the form ... 

Uniform Distribution n A probability distribution where the probability in the case of a discrete random variable or the density function in the case of a continuous random variable, X, with values, x, are constant (equal) over an interval, a,b), where x is greater than or equal to a and X less than or equal to b and x is zero outside the interval. The uniform distribution is sometimes referred to as the rectangular distribution since, a plot of its probability or density function resembles a rectangle. When the random variable, X, is discrete, the uniform distribution is referred to as the discrete uniform distribution and has the probability, P(x. ), with the form of ... 

This discrete random variable distribution is used in circumstances where one is concerned with the probabilities of outcome such as the number of occurrences (e.g., failures) in a sequence of m trials. More specifically, each frial has two possible outcomes (e.g., success or failure), buf fhe probabilify of each trial remains unchanged or constant. 

Figure 7.2 (a) Probability function and (b) cumulative distribution function for discrete random variable. 

Definition Probabihty distribntion of a discrete random variable Let Abe a random variable that may take one of a countable number Mof discrete values X, X2,..., Xm- Let us conduct some very large number Pof trials in which we measure file value of A. Let N(Xj) be the number of times that we observe the value Xj, Ejlj A(A ) = T. Then, the probability distribution of A is defined in file frequentist manner for very large T... 

We have defined the expectation for both discrete and continuous probability distributions that gives the average, or mean value, of a random variable. To obtain a measure ofthe breadth of the distribution ofX, we define the variance to be the expected quadratic variation from the mean,... 

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 ... 

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