91Ó°ÊÓ

Marginal distributions tell us the probability distribution of a single random variable within a joint distribution.
To find the marginal distribution of a variable, we integrate the joint probability density function (PDF) over the range of the other variable.

In our exercise, we have two random variables, X and Y, with a joint PDF proportional to \(x y (x-y)^2\).
To find the marginal distribution of X, we integrate the joint PDF over the entire range of Y:
\[ f_X(x) = \int_{0}^{1} k x y (x-y)^2 \, dy \]

Similarly, to find the marginal distribution of Y, we integrate the joint PDF over the entire range of X:
\[ f_Y(y) = \int_{0}^{1} k x y (x-y)^2 \, dx \]
These integrations simplify the joint PDF to individual PDFs for X and Y.

The correlation coefficient, denoted as \(\rho\), measures the strength and direction of the linear relationship between two random variables.

It takes values between -1 and 1:

• \(\rho = 1\) indicates a perfect positive linear relationship.
• \(\rho = 0\) indicates no linear relationship.
• \(\rho = -1\) indicates a perfect negative linear relationship.

The formula for the correlation coefficient is:
\[ \rho_{XY} = \frac{Cov(X,Y)}{\sqrt{Var(X) Var(Y)}} \]
In this exercise, we are asked to show that X and Y are negatively correlated with \(\rho = -\frac{2}{3}\).
To do so, we need to calculate the covariance and variances of X and Y, and then use them in the formula.

Expected values give us the mean or average value of a random variable over numerous trials.

For a continuous random variable, the expected value is calculated using the integral of the product of the variable and its PDF:
\[ E[X] = \int_{-\infty}^{\infty} x f_X(x) \, dx \]
\[ E[Y] = \int_{-\infty}^{\infty} y f_Y(y) \, dy \]
In our problem, we also need the joint expectation:
\[ E[XY] = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} xy f_{X,Y}(x,y) \, dx \, dy \]
Calculating these expected values provides the necessary information to find variance and covariance.

Covariance indicates the direction of the linear relationship between two random variables.

If both variables tend to increase together, the covariance is positive.
If one tends to increase when the other one decreases, the covariance is negative.
The formula for covariance is:
\[ Cov(X,Y) = E[XY] - E[X]E[Y] \]
In the given problem, after finding \(E[X]\), \(E[Y]\), and \(E[XY]\), we use these values in the covariance formula to determine the linear dependency between X and Y.

A double integral is used to compute the integral of a function over a two-dimensional region.

In this exercise, the double integral is employed to find the constant of proportionality for the joint PDF:
\[ \text{Constant} \times \int_{0}^{1} \int_{0}^{1} x y (x-y)^2 \, dy \, dx = 1 \]
Solving this double integral helps normalize the joint PDF so that the total probability is 1.
Breaking it down:

• First, integrate with respect to y across its range.
• Then, integrate the resulting function with respect to x across its range.

This procedure guarantees that we correctly normalize the joint distribution function.