Question

Let \(X\) and \(Y\) have the joint pdf \(f(x, y)=3 x, 0

Short answer

To find if \(X\) and \(Y\) are independent, we would test if the joint pdf equals to the product of the marginal pdfs. If not, we can then find the conditional expectation \(E(X \mid y)\) by integrating \(x \cdot f_{X|Y}(x|y) \, dx\) over all possible values of \(X\)

Step-by-step solution

Determine Independence

Calculate the marginal pdfs of \(X\) and \(Y\) from the joint pdf \(f(x, y)=3x\), for \(0<y<x<1\) and then check if the joint pdf equals the product of the marginal pdfs. The marginal pdf of \(Y\), denoted as \(f_Y(y)\), can be obtained by integrating the joint pdf over all possible values of \(X\). Similarly, the marginal pdf of \(X\), \(f_X(x)\), can be obtained by integrating the joint pdf over all possible values of \(Y\). Subsequently, if \(f(x, y) = f_X(x)f_Y(y)\), for all \(x\) and \(y\), then the variables are independent.

Calculating E(X|y)

Assuming that the variables are not independent, we need to calculate \(E(X \mid y)\), the expected value of \(X\) given \(Y = y\). This is accomplished by integrating \(x \cdot f_{X|Y}(x|y) \, dx\) over all possible values of \(X\). Where \(f_{X|Y}(x|y)\) is the conditional pdf of \(X\) given \(Y = y\), and can be found by dividing the joint pdf \(f(x, y)\) by the marginal pdf of \(Y\), \(f_Y(y)\).

Stip 3: Solve and Interpret

Finally, the solutions derived from the above steps are to be interpreted. If the joint pdf is equal to the product of the marginal pdfs for all values of \(x\) and \(y\), \(X\) and \(Y\) are independent. If not, the calculated \(E(X \mid y)\) would provide the expected value of \(X\) given \(Y = y\)