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The expected value, often denoted as \(E(X)\) for a random variable \(X\), helps us understand the average, or mean, outcome we can expect. In probability, it is calculated by multiplying each possible value the variable can take by its probability, and then summing all those products. For a discrete random variable, this is mathematically represented as:\[ E(X) = \sum x_i P(X = x_i) \]Where \(x_i\) are the possible values of \(X\), and \(P(X = x_i)\) is the probability of \(X\) being \(x_i\). For example, if \(X\) is our random variable with outcomes from the exercise, calculating \(E(X)\) requires identifying all unique \(x\) values like 0, 1, and 2, with their associated probabilities, and solving for the mean. It's a central measure in probability and statistics, guiding decisions and predictions.

Variance is a core concept that measures how much a set of values are spread out or differ from the expected value. It's denoted as \(V(X)\) for a random variable \(X\). The variance gives insights into the reliability and risk associated with a random process.Mathematically, variance is defined as:\[ V(X) = E(X^2) - [E(X)]^2 \]Here, \(E(X^2)\) is the expected value of the square of \(X\). It represents the mean of the squares of each possible outcome minus the square of the expected mean of outcomes. Higher variance indicates more spread out data, often implying higher risk in practical scenarios, like financial investments. In the exercise, computing variance for \(X\) or \(Y\) involves calculating \(E(X^2)\) or \(E(Y^2)\) by considering the distribution of the squares of outcomes.

Conditional probability distribution refers to the probability of an event occurring given that another event has already occurred. It's a refinement of the idea of probability to situations where certain conditions or "givens" are known.For two random variables \(X\) and \(Y\), the conditional probability of \(Y\) given \(X\) is written as \(P(Y | X)\). This is calculated as:\[ P(Y = y | X = x) = \frac{P(X = x, Y = y)}{P(X = x)} \]In this notion, we focus on the narrowed-down set of possibilities when another event (here \(X\) being a specific value) has occurred. From the solution, given \(X=1\), the conditional distributions of \(Y\) are calculated by focusing on those cases where \(X\) is indeed 1 and normalizing the probabilities of different \(Y\) values in that context.

Independence between two random variables \(X\) and \(Y\) implies that knowing the value of one does not reveal any information about the other. For independence, the joint probability of \(X\) and \(Y\) can be expressed as the product of their individual (marginal) probabilities.Mathematically, \(X\) and \(Y\) are independent if:\[ P(X = x, Y = y) = P(X = x) P(Y = y) \text{ for all } x, y \]In this context, all pairs of values must satisfy this equality for complete independence. The exercise involves checking this condition for each combination of \(X\) and \(Y\). Any deviation from the equality implies dependency, indicating the behavior of one variable affects the other.

The correlation coefficient, usually denoted \( ho\), measures the strength and direction of the linear relationship between two random variables. It's a standardized measure that ranges from -1 to 1, where 1 indicates perfect positive correlation, -1 indicates perfect negative correlation, and 0 means no linear correlation.The formula for correlation is:\[ \rho = \frac{E(XY) - E(X)E(Y)}{\sqrt{V(X)V(Y)}} \]Here, \(E(XY)\) represents the expected value of the product of \(X\) and \(Y\). In the exercise, once \(E(XY)\), \(E(X)\), \(E(Y)\), \(V(X)\), and \(V(Y)\) are computed, \( ho\) can help quantify how \(X\) and \(Y\) move together, indicating how changes in one might be associated with changes in the other.