91Ó°ÊÓ

The expected value, or mean, of a random variable helps us determine the central tendency of a probability distribution. For a random variable \(X\), its expected value is found by integrating the product of the variable and its probability density function across all possible values.
In our exercise, the expected value \(E[X]\) is calculated as \( \int_{0}^{1} x f_X(x) \, dx \), where \(f_X(x)\) is the marginal density function of \(X\). Similarly, for \(Y\), it is \( \int_{0}^{1} y f_Y(y) \, dy \).
The calculations yield \(E[X] = \frac{1}{2}\) and \(E[Y] = \frac{2}{3}\). These values represent the average or expected position of each variable along their respective axes within the defined unit square \([0,1]\times[0,1])\). Identifying the expected values is fundamental for summarizing data properties and making predictions.

Variance is a measure of the spread or dispersion of a set of values. It provides insights into how much the values of a random variable differ from its mean. The variance for a random variable \(X\) is defined as \(\operatorname{Var}(X) = E[X^2] - (E[X])^2\).
The first term, \(E[X^2]\), involves integration of \(x^2\) against its probability density function, evaluating \( \int_{0}^{1} x^2 f_X(x)\, dx \). Similarly for \(Y\), it's \( \int_{0}^{1} y^2 f_Y(y)\, dy \).
Our solution computes \(\operatorname{Var}(X) = \frac{7}{20}\) and \(\operatorname{Var}(Y) = \frac{1}{18}\), indicating how much each random variable differs from its expected value, thus providing an understanding of the variability in the observed behaviors of \(X\) and \(Y\). Both variances are relatively small, indicating that most of the distribution lies close to the expected value.

A marginal density function involves integrating out other variables from a joint density function to isolate the distribution of a single variable. This process simplifies the joint density into a single variate distribution, thereby providing insights into the behavior of individual random variables separately.
In our example, the marginal density of \(X\), \( f_X(x) \), is obtained by integrating the joint density \( f(x, y) \) over \( y \). Conversely, \( f_Y(y) \) is derived by integrating over \( x \).
The steps involve:

• Calculating \( f_X(x) = \int_{0}^{1} 12xy(1-x) \, dy \), resulting in \( 6x(1-x)\).
• Calculating \( f_Y(y) = \int_{0}^{1} 12xy(1-x) \, dx \), resulting in \( 2y \).

The obtained marginal densities \( f_X(x) \) and \( f_Y(y) \) help conclude independence by verifying \( f(x,y) = f_X(x) f_Y(y) \). Furthermore, they are critical for computing the expected values and variances of \(X\) and \(Y\).