Step 1

Here, X and Y are independent, continuous random variables.

So, it can be written as: \(f(x,y) = f(x) f(y)\)

It is given that,

\(f_{X}(x)=\begin{cases}\frac{2}{x^{2}},\ x>2\\0,\ otherwise \end{cases}\)

\(f_{Y}(y)=\begin{cases}\frac{2}{y^{2}},\ y>2\\0,\ otherwise \end{cases}\)

Step 2

(a) The joint probability density function can be obtained as:

\(f_{XY}(x,y)=f_{X}(x)f_{Y}(y)\)

\(=\frac{2}{x^{2}}\times \frac{2}{y^{2}}\)

\(=\frac{4}{(xY)^{2}}\)

Thus,

\(f_{XY}(x,y)=\begin{cases}\frac{4}{(xy)^{2}},\ x>2, y>2\\0,\ otherwise\end{cases}\)

Step 3

(b)The probability density function of \(W=XY\) is same as probability density function of \(f(x,y).\)

\(f_{W}(w)=f_{XY}(x,y)=\begin{cases}\frac{4}{(xy)^{2}},\ x>2, y>2\\0, \ otherwise\end{cases}\)