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In probability theory, a joint probability distribution describes the likelihood of two or more random variables occurring simultaneously. For this exercise, we are interested in finding the joint probability distribution of two variables: \(X\) and \(Y\), where \(X\) represents the number of sampled pages with moderate graphics, and \(Y\) represents those with high graphics.
Here, the joint probability distribution \(f_{XY}(x, y)\) is derived using the hypergeometric distribution. This distribution is suitable because we are sampling without replacement from a finite population. The population consists of 100 web pages, out of which 30 have moderate graphics and 10 have high graphics.
The joint probability function is given by:

• \[ f_{XY}(x, y) = \frac{\binom{30}{x} \binom{10}{y} \binom{60}{4-x-y}}{\binom{100}{4}} \]

This formula calculates the probability of selecting \(x\) pages with moderate graphics and \(y\) with high graphics, ensuring that their sum along with any pages with low graphics equals our total sample size of 4. The variables \(x\) and \(y\) must satisfy \(x + y \leq 4\) and \(x, y \geq 0\).
This joint distribution reflects all combinations of moderate and high graphics possibilities in the sample, providing a comprehensive view of their probabilities.

Marginal probability distribution is the probability distribution of a subset of variables within a larger set when the remaining variables are not specified or considered. In this exercise, we focus on finding the marginal probability distribution of the variable \(X\), the number of pages with moderate graphics.
To derive the marginal probability distribution \(f_X(x)\), we sum the joint probability distribution over all possible values of \(Y\). This involves considering all feasible values of \(y\) that can be paired with \(x\) such that their sum does not exceed the sample size. For each specific \(x\), we calculate:

• \[ f_X(x) = \sum_{y=0}^{4-x} f_{XY}(x, y) \]

By performing this summation, we extract the probabilities associated solely with \(x\), the moderate graphics pages, effectively "marginalizing" over \(y\).
This marginal distribution helps us understand the likelihood of observing a given number of moderate graphics pages within our sample, irrespective of the number of high graphics pages included.

Expectation in the context of probability, often referred to as the expected value or mean, represents the average outcome of a random variable if the experiment were repeated many times. For a hypergeometric distribution, which applies to our scenario of sampling without replacement, the expected value can be calculated using a simple formula.
For our variable \(X\), the expected number of pages with moderate graphics is determined with:

• \[ E(X) = n \times \frac{K}{N} = 4 \times \frac{30}{100} = 1.2 \]

Here,

• \(n\) is the sample size, which is 4.
• \(K\) is the number of pages with moderate graphics in the entire website, which is 30.
• \(N\) is the total number of pages, which is 100.

Thus, the expected value \(E(X)\) equals 1.2, indicating that on average, one can anticipate selecting about 1.2 pages with moderate graphics in a random sample of four pages.
Expectation gives us an idea of the central tendency of a probability distribution and is a crucial concept in both theoretical and applied probability.