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Prompt:

Find examples in economics where the concept of conditional probability is relevant.

Find examples in economics for which discrete distributions arise.

Lesson

In this lesson, we introduce the concept of a random variable (r.v.). A random variable maps some chance outcome to a value on the real line. For discrete r.v.s, many events are relatively straightforward to conceptualize – coin tosses, choosing cards from a deck. Indeed, gambling served as an initial motivator in the development of probability. When the events come from a sample space which is a portion of or the entire real line, we need to be careful in how we define continuous r.v.s. We will introduce bivariate random variables, that is, pairs of r.v.s that are probabilistically related to one another. In this context, we want to ask the questions, what is the distribution one variable when we have summed or integrated the probability of the other random variable across all events, and what is the distribution of one variable if the other variable is held constant? Finally, we introduce expectation and how it is calculated for discrete and continuous random variables. Chapter 3 can be a bit daunting-looking because of the number of integrals you see. I won’t have you do any real integration, so you don’t need to brush off your calculus book to find out how to do integration by parts or integration by substitution. What I do want you to understand are the underlying concepts that are represented. For example, if a pair of random variables has a bivariate distribution, then to compute the marginal distribution for one random variable, you integrate out the other random variable. In essence, what you are doing is taking a weighted average of the random variable you’re interested in with the weights the probabilities of the variable you are not interested in.

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