# Marginal density from joint distribution of 3 random variables

Suppose random variables X, Y, Z have joint probability density function given as $$f_{XYZ}(x,y,z) = \frac{2x}{(A_2-A_1)(R_2^{2} - R_1^{2})y^{2}z} e^{-z^{2}/2}$$ where A1,A2,R1,R2 are constant. How do I find the marginal probability density function of X, if the support is \begin{cases}\ R_1yz\leq x \leq R_2yz \\ A_1 \leq y \leq A_2 \\ 0 \leq z < \infty & \end{cases} I know that I need to integrate with respect to Y (the will be two regions, I guess) and Z to find marginal P.D.F. of X. But I don't know what the boundary of the integrals should be, especially I cant understand how to deal with the range of Z ($\infty$ is causing the main problem). What will be the final range of PDF of X. Thanks in advance.

• What is the joint density? What are $A_1$ and $A_2$? By $\inf$ do you mean $\infty$? – Math1000 Jan 12 '15 at 0:25
• @ Math 1000, I have edited the question. Please have a look. – PD007 Jan 12 '15 at 6:08