To find the joint density function $f_{X,Y}(x,y)$ of the random variables $X$ and $Y$ that are uniformly distributed over the region $D$, we need to understand the region $D$ defined by the conditions:

• $3X > Y$
• $0 < x < 1$
• $y > 0$

The region $D$ describes the set of all points $(x, y)$ such that:

• $x$ ranges from 0 to 1,
• $y$ ranges from 0 to $3x$.

Given that $(X, Y)$ is uniformly distributed over this region, the joint density function $f_{X,Y}(x,y)$ will be constant in this region. The area of the region $D$ needs to be calculated to find the constant value of the density function.

Calculating the area of $D$:

To find the area, we integrate the height $y = 3x$ over the interval $x = 0$ to $x = 1$:

$\text{Area} = \int_{0}^{1} 3x \, dx$

$\text{Area} = 3 \left[\frac{x^2}{2}\right]_{0}^{1} = 3 \cdot \frac{1}{2} = \frac{3}{2}$

Since the density function must integrate to 1 over the region, the value of the density function is the reciprocal of the area:

$f_{X,Y}(x,y) = \frac{1}{\text{Area}} = \frac{1}{\frac{3}{2}} = \frac{2}{3}$

Thus, the joint density function is:

\frac{2}{3} & \text{for } 0 < x < 1, 0 < y < 3x \\ 0 & \text{otherwise} \end{cases}$$ Would you like details or have any questions? Here are 8 related questions you might find interesting: 1. How do you find the marginal density functions from the joint density function? 2. What is the expected value of $$X$$ and $$Y$$ given their joint density function? 3. How do you compute the covariance between $$X$$ and $$Y$$ for this joint distribution? 4. What is the conditional density function of $$Y$$ given $$X$$? 5. How would you find the probability that $$X$$ is less than a certain value? 6. Can you explain what it means for two random variables to be independent? 7. What are the implications of a uniform distribution over a certain region? 8. How can you verify if a joint density function is valid? **Tip:** When dealing with joint density functions, always ensure that the total probability integrates to 1 over the defined region. This is a key property of any probability density function.