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In probability and statistics, a random variable is a fundamental concept that simplifies complex real-world scenarios into understandable numeric outcomes. It's a variable whose possible values are numerical outcomes of random phenomena. For instance, in the context of the raffle tickets scenario, the random variable \(X\) is defined as your winnings when purchasing four tickets. The variable \(X\) is "random" because it can yield several possible outcomes depending on the event (whether you win or not).
The values \(X\) can adopt are determined by the number of tickets purchased relative to the total tickets sold and whether they result in a winning ticket. Here, your winnings if you purchase four raffle tickets can be either \(\\(0\) (if none of your tickets win) or \(\\)150\) (if one of the tickets wins the prize). Defining \(X\) helps in analyzing the situation by reducing it to numeric values that can be worked with to calculate probabilities and expectations.

Expected value essentially provides you with an average outcome of a random variable if you were to repeat an experiment multiple times. It combines results and their corresponding probabilities into a single metric.
In the case of the raffle tickets, the expected value \(E(X)\) of your winnings is a figure that indicates what you can expect, on average, to win from your tickets. This is computed by multiplying each possible outcome by its probability and summing these products together. For example, if you win \(\\(150\) with a probability of \(0.04\) and win \(\\)0\) with a probability of \(0.96\), the expected value is calculated as follows:

• \(E(X) = (150 \times 0.04) + (0 \times 0.96) = 6\)

This means, over many repetitions of the fund-raiser where you always buy four tickets, your "average" winnings will be \(\$6\) per raffle. Understanding expected value is essential because it gives insight into the profitability or expected gain/loss of an event over the long term.

A Probability Density Function (PDF) is a tool that describes the likelihood of a random variable assuming particular values. It's a way to represent the distribution of probabilities for all possible outcomes of a random variable.
Let's consider the raffle ticket example. Here, the PDF is used to illustrate the probabilities associated with your winnings being \(\\(150\) or \(\\)0\). Specifically, it is noted in the PDF that:

• \(P(X = 150) = 0.04\)
• \(P(X = 0) = 0.96\)

This shows how likely each outcome is, thus enabling you to visualize and make informed decisions about the situation in question. In this raffle case, the PDF conveys that although there's a small chance for a large win, the probability of not winning (and thus receiving \(\$0\)) is quite high. PDFs are indispensable in both academic studies and practical applications involving uncertainty and risks.