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The expected value, often called the mean, is a fundamental concept in probability and statistics. It provides a measure of the central tendency of a random variable. In simpler terms, the expected value is the average outcome you'd expect if you were to repeat an experiment infinitely many times.

To compute the expected value for continuous random variables like in this exercise, we use integration. For a random variable \( Y_1 \), its expected value \( E(Y_1) \) is calculated by integrating the product of the variable and its probability density function over its range:\[E(Y_1) = \int_{-fty}^{fty} y_1 f(y_1) \, dy_1\]where \( f(y_1) \) is the marginal probability density function of \( Y_1 \).

In our solution, we first find the marginal probability density functions before calculating \( E(Y_1) \) and \( E(Y_2) \). This process helps us to accurately understand the behavior of each variable independently.

Variance is a measure of how much a random variable's values deviate from their expected value. It quantifies the spread of a distribution. A larger variance indicates that values are more spread out; a smaller variance means they are closer to the mean. Variance is crucial when assessing the reliability and predictability of random outcomes.

Mathematically, variance of a random variable \( Y_1 \) is defined as:\[V(Y_1) = E(Y_1^2) - (E(Y_1))^2\]The term \( E(Y_1^2) \) is the expected value of the square of the random variable, and \( (E(Y_1))^2 \) is the square of the expected value of the random variable.

In the solved exercise, we determined \( V(Y_1) \) and \( V(Y_2) \) by first finding \( E(Y_1^2) \) and \( E(Y_2^2) \) through integration and then applying the variance formula. The calculations reveal how scattered or concentrated the values can be around the mean.

The marginal probability density function (PDF) describes the probability distribution of a subset of a collection of random variables, disregarding the presence of other variables. In simpler terms, it provides the probability density of a single variable within a multi-variable distribution.

To extract the marginal PDF of a variable, you integrate the joint PDF over the range of the other variable(s). For instance, if you have two variables, \( Y_1 \) and \( Y_2 \), to find the marginal PDF of \( Y_1 \), you integrate the joint PDF \( f(y_1, y_2) \) with respect to \( y_2 \):\[f_{Y_1}(y_1) = \int_{y_1}^{1} f(y_1, y_2) \, dy_2\]This results in a function dependent only on \( y_1 \).

In this exercise, we calculated both \( f_{Y_1}(y_1) \) and \( f_{Y_2}(y_2) \) to help isolate the behaviors of \( Y_1 \) and \( Y_2 \) independently.

Linearity of expectation is a very useful property in probability theory, particularly because it holds regardless of whether the random variables are independent. This property states that the expected value of a sum of random variables is equal to the sum of their expected values. Formally, for any two random variables \( X \) and \( Y \):\[E(X + Y) = E(X) + E(Y)\]This can be extended to:\[E(aX + bY) = aE(X) + bE(Y)\]where \( a \) and \( b \) are constants.

For the given exercise, linearity of expectation was used to calculate \( E(Y_1 - 3Y_2) \). Instead of needing complex calculations for this combination, we simply took the expected value of \( Y_1 \) and subtracted three times the expected value of \( Y_2 \). This significantly simplifies the problem-solving process and demonstrates the power of this concept.