We start our investigation of random variables. Descriptive statistics deals with variables that can take values within a discrete or a continuous set. Correspondingly, we cover discrete random variables. As we shall see, the mathematics involved in the study of continuous random variables requires concepts from calculus and is a bit more challenging than what… Continue reading Introduction
Month: February 2023
TOTAL PROBABILITY AND BAYES’ THEOREMS
Conditional probabilities are a very important and powerful concept. In this section we see how we may tackle problems like the one in Example 5.2, which we use as a guideline. To frame the problem clearly, let us define the following events: Now the first question is: What do we know and what would we like… Continue reading TOTAL PROBABILITY AND BAYES’ THEOREMS
CONDITIONAL PROBABILITY AND INDEPENDENCE
Consider throwing a die twice. If we know that the result of the first draw is 4, does this change our probability assessment for the second draw? If the die is fair, and there is no cheating on the part of the person throwing it, the answer should be no. The two rolls are independent.… Continue reading CONDITIONAL PROBABILITY AND INDEPENDENCE
Probability measures
The final step is associating each event E ∈ F with a probability measure P(E), in some sensible way. As a starting point, it stands to reason that, for an event E ⊆ Ω, its probability measure should be a number satisfying the following condition: This is certainly true if we think of probabilities in terms of relative frequencies, but it… Continue reading Probability measures
The algebra of events
Given the definition of events, let us consider how we may build possibly complex events that have a practical relevance. Indeed, we often deal with the following concepts: Since events are sets, it is natural to translate the concepts above in terms of set theory, relying on the usual difference, union, and intersection of sets.… Continue reading The algebra of events
THE AXIOMATIC APPROACH
The axiomatic approach aims at building a consistent theory of probability and is based on the following logical steps: 5.2.1 Sample space and events To get going, we should first formalize a few concepts about running a random experiment and observing outcomes. The set of possible outcomes is called the sample space, denoted by Ω. For… Continue reading THE AXIOMATIC APPROACH
DIFFERENT CONCEPTS OF PROBABILITY
We have met relative frequencies, a fundamental concept in descriptive statistics. Intuitively, relative frequencies can be interpreted as “probabilities” in some sense, as they should tell us something about the likelihood of events. While this is legitimate and quite sensible in many settings, we should wonder whether this frequentist interpretation is the only meaning that we may… Continue reading DIFFERENT CONCEPTS OF PROBABILITY
Introduction
The probability theory, and this one is no exception. However, the careful reader should wonder title mentions probability theories. In Section 5.1 we show that probability, like uncertainty, is a rather elusive concept. Descriptive statistics suggests the concept of probabilities as relative frequencies, but we may also interpret probability as plausibility related to a state of belief. The… Continue reading Introduction
MULTIDIMENSIONAL DATA
So far, we have considered the organization and representation of data in one dimension, but in applications we often observe multidimensional data. Of course, we may list summary measures for each single variable, but this would miss an important point: the relationship between different variables. In issues concerning independence, correlation, etc. Here we want to… Continue reading MULTIDIMENSIONAL DATA
Quartiles and boxplots
Among the many percentiles, a particular role is played by the quartiles, denoted by Q1, Q2, and Q3, corresponding to 25%, 50%, and 75%, respectively. Clearly, Q2 is simply the median. A look at these values and the mean tells a lot about the underlying distribution. Indeed, the interquartile range has been proposed as a measure of dispersion, and an alternative measure… Continue reading Quartiles and boxplots