Author: haroonkhan
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EXPECTED VALUE
Both PMF and CDF provide us with all of the relevant information about a discrete random variable, maybe too much. In descriptive statistics, we use summary measures, such as mean, median, mode, variance, and standard deviation, to get a feeling for some essential features of a distribution, like its location and dispersion. In probability theory,…
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Probability mass function
The CDF looks like a somewhat weird way of describing the distribution of a random variable. A more natural idea is just assigning a probability to each possible outcome in the support. Unfortunately, in the next chapter we will see that this idea cannot be applied to a continuous random variable. Nevertheless, in the case…
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Cumulative distribution function
The basic stuff of probability theory consists of events and their probability measures. Given a random variable X, consider the event {X ≤ x}; incidentally, note how we use x to denote a number. The probability of this event is a function of x. DEFINITION 6.3 (Cumulative distribution function) Let X be a random variable. The function for , is called cumulative distribution function (CDF). The notation FX(x) clarifies…
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CHARACTERIZING DISCRETE DISTRIBUTIONS
When we deal with sampled data, it is customary to plot a histogram of relative frequencies in order to figure out how the data are distributed. When we consider a discrete random variable, we may use more or less the same concepts in order to provide a full characterization of uncertainty. For instance, if we consider…
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RANDOM VARIABLES
In probability theory we work with events. The questions we may ask about events are quite limited, as they can either occur or not, and we may just investigate the probability of an event. In business management, more often than not we are interested in questions with a more quantitative twist, since events are linked…
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Introduction
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…
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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…
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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.…
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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…
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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.…