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Probability is a universally accepted tool for expressing degrees of confidence or doubt about some proposition in the presence of incomplete information or uncertainty. By convention, probabilities are calibrated on a scale of 0 to 1; assigning something a zero probability amounts to expressing the belief that we consider it impossible, while assigning a probability of one amounts to considering...
In this chapter, we offer a glimpse into some problems that have earned the status of being classics in counting and combinatorial probability. They have an entertainment value, and they also present some surprises in their solutions and the final answers. The problems we present are generally known as the birthday problem and the matching problem. For greater exposure to the material in this chapter,...
Both conditional probability and independence are fundamental concepts for probabilists and statisticians alike. Conditional probabilities correspond to updating one’s beliefs when new information becomes available, a natural human instinct. Independence corresponds to irrelevance of a piece of new information, even when it is made available. Additionally, the assumption of independence can and does...
In this chapter, we introduce the concept of random variables and their distributions. In some sense, the entire subject of probability and statistics is about distributions of random variables. Random variables, as the very name suggests, are quantities that vary over time or from individual to individual, and the reason for the variability is some underlying random process. We try to understand...
Studying distributions of random variables and their basic quantitative properties, such as expressions for moments, occupies a central role in both statistics and probability. It turns out that a function called the probability generating function is often a very useful mathematical tool in studying distributions of random variables. It is useful to derive formulas for moments and for the pmf of...
A few special discrete distributions arise very frequently in applications. Either the underlying probability mechanism of a problem is such that one of these distributions is truly the correct distribution for that problem or the problem may be such that one of these distributions is a very good choice to model that problem. We present these distributions and study their basic properties in this...
We mentioned in Chapter 4 that discrete random variables serve as good examples to develop probabilistic intuition, but they do not account for all the random variables that one studies in theory and applications. In this chapter, we introduce the so-called continuous random variables, which typically take all values in some nonempty interval; e.g., the unit interval, the entire real line, etc. The...
A number of densities, by virtue of their popularity in modeling or because of their special theoretical properties, are considered to be special. In this chapter, we present a collection of these densities with their basic properties.We discuss, when suitable, their moments, the form of the CDF, the mgf, shape and modal properties, and interesting inequalities. Classic references to standard continuous...
Empirical data on many types of variables across disciplines tend to exhibit unimodality and only a small amount of skewness. It is quite common to use a normal distribution as a model for such data. The normal distribution occupies the central place among all distributions in probability and statistics. When a new methodology is presented, it is usually first tested on the normal distribution. The...
Many of the special discrete and special continuous distributions that we have discussed can be well approximated by a normal distribution for suitable configurations of their underlying parameters. Typically, the normal approximation works well when the parameter values are such that the skewness of the distribution is small. For example, binomial distributions are well approximated by a normal distribution...
We have provided a detailed treatment of distributions of one discrete or one continuous random variable in the previous chapters. But often in applications we are just naturally interested in two or more random variables simultaneously. We may be interested in them simultaneously because they provide information about each other or because they arise simultaneously as part of the data in some scientific...
Similar to the case of several discrete random variables, in applications we are frequently interested in studying several continuous random variables simultaneously. An example would be a physician’s measurement of a patient’s height, weight, blood pressure, electrolytes, and blood sugar. Analogous to the case of one continuous random variable, again we do not talk of pmfs of several continuous variables...
Very naturally, in applications we often want to study suitable functions or transformations of an original collection of variables X1, X2,…, Xn. For example, the original variables X1, X2,…, Xn could be the inputs into some process or system, and we may be interested in the output, which is some suitable function of these input variables...
In many applications, successive observations of a process, say X1, X2,…, have an inherent time component associated with them. For example, the Xi could be the state of the weather at a particular location on the i th day, counting from some fixed day. In a simplistic model, the state of the weather could be “dry” or “wet,” quantified as,...
Urn models conceptualize general allocation problems in which we distribute, withdraw, and redistribute certain objects or units into a specified number of categories. We think of the categories as urns and the objects as balls. Depending on the specific urn model, the balls may be of different colors and distinguishable or indistinguishable. Urn models are special because they can be successfully...
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