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When we talk about a probability distribution, we're referring to a mathematical function that provides the probabilities of occurrence of different possible outcomes for an experiment. It's like a map that tells us how likely each outcome of a random event is. Each value that a random variable can take has an associated probability that must be between 0 and 1, and the sum of all probabilities must equal 1.

Let's take a lottery as an example. If you have a lottery ticket, the probability distribution would tell you the chance of winning different prize amounts. The ticket could win nothing, a small prize, or the grand prize, and there are different probabilities for each of these outcomes. In our exercise, the distribution is given in a table format, showing the different values that the variable X can take, such as -5, -1, 0, 1, 5, and 8, and the corresponding probabilities for X to take on those values.

A random variable is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types of random variables: discrete and continuous. Discrete random variables have a countable number of possible values, like the number of books in a backpack, while continuous random variables have an infinite number of possibilities within a range, like the amount of milk in a jug.

In our exercise scenario, we're dealing with a discrete random variable. The variable X represents the outcome of some process, and we know the exact values it can take. As seen in the exercise, X can be -5, -1, 0, 1, 5, or 8, each with an assigned probability. The idea behind a random variable is that we don't know which value will occur on a given trial; we only know how likely each value is.

The mathematical expectation, also known as the expected value, is a key concept in probability and statistics. It represents the average value of a random variable over a large number of trials or occurrences. Think of it as the long-term average if you were to repeat an experiment over and over again. The expected value is found by multiplying each possible value of the variable by its probability and then summing all these products.

For example, if you were flipping a fair coin, the expected value of getting heads would be 0.5 since there's a 50% chance of getting heads on any flip. But expectations aren't just for simple events; they're crucial in more complex scenarios such as gambling, insurance policies, and stock market investments. In our exercise, step by step, we computed the expected value by following this same logic, leading us to conclude that the expected value of X is 0.86. This doesn't mean we'll get exactly 0.86 in one trial, but rather that 0.86 is the average outcome if we could repeat the experiment an infinite number of times.