Discrete random variables, probability

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

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. 

In tlie case of a discrete random variable, tlie cdf is a step function increasing by finite jumps at die values of x in die range of X. In die example above, diese jumps occur at die values 2, 5, and 7. Tlie magnitude of each jump is equal to die probability assigned to die value ii liere the jmnp occurs. Tliis is depicted in Fig. 19.8.3. Another form of representing die cdf of a discrete random variable is provided in Figure 19.8.4. 

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... 

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. 

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. 

Probability Generating Function. For a discrete random variable, x, the function... 

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. 

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 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 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. 

Probability density function (pdf) Indicates the relative likelihood of the different possible values of a random variable. For a discrete random variable, say X, the pdf is a function, say /, such that for any value x, /(x) is the probability that X = X. For example, if X is the number of pesticide applications in a year, then /(2) is the probability density function at 2 and equals the probability that there are two pesticide applications in a year. For a continuous random variable, say Y, the pdf is a function, say g, such that for any value y, g(y) is the relative likelihood that Y = y,0 < g y), and the integral of g over the range of y from minus infinity to plus infinity equals 1. For example, if Y is body weight, then g(70) is the probability density function for a body weight of 70 and the relative likelihood that the body weight is 70. Furthermore, if g 70)/g(60) = 2, then the body weight is twice as likely to be 70 as it is to be 60 (Sielken, Ch. 8). 

The probability density plays the same role for continuous variables as does P(X) for discrete random variables. The normalization condition becomes... 

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 probabilities of values of the random variable occurring must sum to 1 (in the case of a discrete random variable) or integrate to 1 (in the case of a continuous random variable). 

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. 

One of the most useful features of decision trees is their ability to illustrate variable outcomes when their probabilities of occurrence can be estimated. To be effective in a graphic format, the outcomes must be characterized as discrete random variables with a relatively small number of possibilities. 

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. 

The probability function for a discrete random variable is sometimes known as a probability mass Junction. The equivalent function for a continuous random variable is a probability density Junction (pdf)-... 

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... 

Discrete random variables are characterized by a discrete probability density which defines the probability associated with some set of states, that is if we have a random variable X which can take on states i = 1,2,..the non-negative density may be viewed as a sequence f = / which is assumed to be summable with = 1. Then the probability that the random variable lies in a given subset of the natural numbers is just the sum of the corresponding density values ... 

For many problems there are situations involving more than a single random variable. Consider the case of 2 discrete random variables, x and y. Here y may take any of the values of the set +". Now instead of having PMFs of a single variable, the joint PMF,P(x,y) is required. This may be viewed as the probability of x taking a particular values and y taking a particular value. This joint PMF must satisfy... 

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