Variables discrete random

Let some event have have M possible outcomes. We are interested in the probability of each of these outcomes occurring. Let the set of possible outcomes be 

---

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, for e.xample, a box of 100 transistors containing five defectives. Suppose tliat a transistor selected at random is to be classified as defective or non-defective. Let X denote die outcome, widi X = 0 associated with die drawing of a non-defective and X = 1 associated with die drawing of a defective. Then X is a discrete random variable with pdf specified by... 

Figure 19.8.3. Graph of tlie cdf of a discrete random variable X.

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. 

If X is a discrete random variable witli pdf specified by f(x), then ... 

To illustrate tlie computation of variance and its interpretation in the case of discrete random variables, consider a random variable X having pdf specified 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. 

A discrete random variable is one that may take on only distinct, usually integer, values. A continuous random variable is one that may take on any value within a continuum of values. 

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... 

The most important characteristic of self information is that it is a discrete random variable that is, it is a real valued function of a symbol in a discrete ensemble. As a result, it has a distribution function, an average, a variance, and in fact moments of all orders. The average value of self information has such a fundamental importance in information theory that it is given a special symbol, H, and the name entropy. Thus... 

In applications it is often a matter of convenience if random demands are modeled by a continuous or by a discrete random variable. In both cases we require that a random demand can not be negative. In this section we assume continuous demands and assure the reader that all formulas have a straightforward extension to the discrete case. Therefore we usually skip the adjective continuous. 

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. 

Recall from calculus (or physics or statistics) that distributions can have k different moments about the origin, which for a discrete random variable. Mi, are given by... 

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 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. 

Consider the situation in which a chemist randomly samples a bin of pharmaceutical granules by taking n aliquots of equal convenient sizes. Chemical analysis is then performed on each aliquot to determine the concentration (percent by weight) of pseudoephedrine hydrochloride. In this example, measurement of concentration is referred to as a continuous random variable as opposed to a discrete random variable. Discrete random variables include counted or enumerated items like the roll of a pair of dice. In chemistry we are interested primarily in the measurement of continuous properties and limit our discussion to continuous random variables. 

Discrete random variable A random variable that can take on only a countable number of different values, e.g., the number of children in a household is a discrete random variable (Sielken, Ch. 8). 

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). 

Let X denote die number of die tlirow on wliich die first failure of a switch occurs. Then X is a discrete random variable widi range 1, 2, 3,. .., n,. ... Note duit die nuige of X consists of a countable infinitude of values and diat X is dierefore discrete. 

---