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When studying a certain phenomenon like how often students visit a dentist, we encounter a set of values each with a particular likelihood of occurring. This is known as a probability distribution, and it's essentially a mathematical description of outcomes. For a discrete random variable, such as the number of dental visits by college students in a year, a probability distribution assigns a probability to each possible value.

Picture this: Every outcome that the random variable can take on is listed, just like in a menu at a restaurant. The probability distribution tells you how 'popular' each menu item is, that is, how likely it is to be chosen. In our dental visit example, the probabilities are 0.15 for 0 visits, 0.50 for 1 visit, and 0.35 for 2 visits. Notice that the probabilities add up to 1, which means we've accounted for all possible outcomes — a fundamental property of probability distributions.

The expected value, often symbolized as \(\mu\) or \(E(X)\), goes by many names — mean, average, first moment. But regardless of the term you use, it serves as a measure of the 'center' of the distribution, giving us an idea of the typical value we expect from our random variable.

So, how to find it? Imagine our dentist visit scenario as a game of chance. If you were to play this game over and over, what would be your average score? To compute the expected value, we take each number of visits, called \(Y\), and weigh it by its probability, \(P(Y)\). Calculating \(\sum (Y \times P(Y))\) gives us the sum of these weighted visits, which in our case resulted in 1.20 visits. This average represents what you might expect over many instances of the scenario.

A discrete random variable is like a box of chocolates with a limited selection — you know exactly what pieces you might get. It can only take on a countable number of values, such as 0, 1, or 2 dentist visits. Each value has its own probability that is greater than or equal to 0 and all probabilities summed together equal 1.

Think of it in terms of a simple coin flip: heads or tails. There are two possible outcomes, making the variable 'discrete'. Now, apply this to our example of dental visits. Here, our discrete random variable is the number of visits, with each possibility (0, 1, or 2) happening with different probabilities. It is an excellent way to handle real-world processes that don't have infinite possibilities but a distinct set of outcomes.