91Ó°ÊÓ

The statement claims that if \(X\) and \(Y\) are independent random variables, then \(1 / X\) and \(1 / Y\) will also be independent. To determine the validity of this statement, let's consider the definition of independence: Two random variables \(X\) and \(Y\) are independent if and only if the joint probability distribution function (pdf) of \(X\) and \(Y\) is the product of the marginal pdfs of \(X\) and \(Y\), i.e., \(f_{XY}(x, y) = f_X(x) f_Y(y)\) for all \(x\) and \(y\). If \(X\) and \(Y\) are independent, then the joint pdf of \(U = 1 / X\) and \(V = 1 / Y\) can be expressed as: $$f_{UV}(u, v) = \frac{\partial(x, y)}{\partial(u, v)} f_{XY}(x, y) = \frac{\partial(x, y)}{\partial(u, v)} f_X(x) f_Y(y).$$ Now we need to find the Jacobian of the transformation, which is given by: $$\frac{\partial(x, y)}{\partial(u, v)} = \frac{\partial(x, y)}{\partial(u)} \cdot \frac{\partial(x, y)}{\partial(v)} = \frac{ \partial (1/u) }{ \partial u } \cdot \frac{ \partial(1/v) }{ \partial v } = (-\frac{1}{u^2})(-\frac{1}{v^2}).$$ Substituting this into the joint pdf of \(U\) and \(V\), we get: $$f_{UV}(u, v) = (-\frac{1}{u^2})(-\frac{1}{v^2})f_X(\frac{1}{u}) f_Y(\frac{1}{v}) = f_U(u) f_V(v).$$ As we can see, the joint pdf of \(U\) and \(V\) is the product of their marginal pdfs, which means that \(X\) and \(Y\) being independent implies that \(1 / X\) and \(1 / Y\) are also independent. The first statement is correct.

The second statement claims that if \(X\) and \(Y\) are uncorrelated random variables, then \(1 / X\) and \(1 / Y\) will also be uncorrelated. Recall that two random variables are uncorrelated if their covariance is zero, i.e., \(Cov(X, Y) = E(XY) - E(X)E(Y) = 0\). Let \(U = 1/X\) and \(V = 1/Y\). Now, we need to find whether \(Cov(U, V) = 0\) or not. Using the definition of covariance, we have: $$Cov(U, V) = E(UV) - E(U)E(V) = E\left(\frac{1}{XY}\right) - E\left(\frac{1}{X}\right)E\left(\frac{1}{Y}\right).$$ It is not guaranteed that \(Cov(U, V) = 0\) just because \(Cov(X, Y) = 0\). Therefore, the second statement is false. If \(X\) and \(Y\) are uncorrelated, it does not necessarily imply that \(1 / X\) and \(1 / Y\) are uncorrelated.