91Ó°ÊÓ

#Step 2: Find the joint density function of X and Y# Since X and Y are independent and uniformly distributed over (0, 1), we have: \(f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y) = 1 \cdot 1 = 1\) for \(0 < x < 1\) and \(0 < y < 1\). #Step 3: Calculate the Jacobian of the transformation matrix# To calculate the Jacobian, we first need to find the partial derivatives of x and y with respect to R and \(\Theta\). %x(R, \Theta) = R \cos(\Theta)\), and \(y(R, \Theta) = R \sin(\Theta)\) \(\frac{\partial x}{\partial R} = \cos(\Theta)\), \(\frac{\partial x}{\partial \Theta} = -R \sin(\Theta)\), \(\frac{\partial y}{\partial R} = \sin(\Theta)\), \(\frac{\partial y}{\partial \Theta} = R \cos(\Theta)\) Now, we can calculate the Jacobian: \(J(R, \Theta) = \begin{vmatrix} \frac{\partial x}{\partial R} & \frac{\partial x}{\partial \Theta} \\ \frac{\partial y}{\partial R} & \frac{\partial y}{\partial \Theta} \end{vmatrix} = \begin{vmatrix} \cos(\Theta) & -R \sin(\Theta) \\ \sin(\Theta) & R \cos(\Theta) \end{vmatrix} = R \cos^2(\Theta) + R \sin^2(\Theta) = R\) #Step 4: Apply the Jacobian transformation formula# Finally, we apply the Jacobian transformation formula to find the joint density function of R and \(\Theta\): \(f_{R,\Theta}(r, \theta) = f_{X,Y}(x(r, \theta), y(r, \theta)) \cdot |J(r, \theta)|\) \(f_{R,\Theta}(r, \theta) = 1 \cdot |r|\), for \(0 < r < \sqrt{2}\) and \(0 < \theta < \frac{\pi}{2}\) Therefore, the joint density function of R and \(\Theta\) is: \(f_{R,\Theta}(r, \theta) = r\), for \(0 < r < \sqrt{2}\) and \(0 < \theta < \frac{\pi}{2}\).