Let (X,Y)∼Uniform(D)(*X*,*Y*)∼Uniform(*D*), where D:={3X>Y,0<x<1,y>0}*D*:={3*X*>*Y*,0<*x*<1,*y*>0}



*1 point*

Find the joint density of X*X* and Y.*Y*.

fXY(x,y)={32for 0<x<1,0<y<3x0otherwise*fXY*​(*x*,*y*)=⎩⎨⎧​23​0​for 0<*x*<1,0<*y*<3*x*otherwise​

fXY(x,y)={23for 0<x<1,0<y<3x0otherwise*fXY*​(*x*,*y*)=⎩⎨⎧​32​0​for 0<*x*<1,0<*y*<3*x*otherwise​

fXY(x,y)={23for 0<x<1,y>3x0otherwise*fXY*​(*x*,*y*)=⎩⎨⎧​32​0​for 0<*x*<1,*y*>3*x*otherwise​

fXY(x,y)={23for 0<x<1,y>00otherwise*fXY*​(*x*,*y*)=⎩⎨⎧​32​0​for 0<*x*<1,*y*>0otherwise

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:

\[
f_{X,Y}(x,y) = \begin{cases} 
\frac{2}{3} & \text{for } 0 < x < 1, 0 < y < 3x \\ 
0 & \text{otherwise} 
\end{cases}
\]

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.

Learn how to calculate the joint density function fXY(x,y) for the uniform distribution of random variables X and Y over a specified region. This problem involves understanding the region D defined by conditions such as 3X > Y, 0 < x < 1, and y > 0. Suitable for advanced undergraduate students in probability theory.

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