Question

Let X and Y be independent, continuous random variables with the same maginal probability density function, defined as f_{X}(t)=f_{Y}(t)=\begin{cases}

Random variables

Let X and Y be independent, continuous random variables with the same maginal probability density function, defined as
$$f_{X}(t)=f_{Y}(t)=\begin{cases}\frac{2}{t^{2}},\ t>2\\0,\ otherwise \end{cases}$$
(a)What is the joint probability density function f(x,y)?
(b)Find the probability density of W=XY. Hind: Determine the cdf of Z.

2021-06-07

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