Finding Joint PDF of Two Exponential Random Variables

[EugP]

Can anyone tell me how to find the joint PDF of two random variables? I can't seem to find an explanation anywhere. I'm trying to solve a problem but I'm not sure where to go with it:

Y is an exponential random variable with parameter $$\lambda=4$$. X is also an exponential random variable and independent of Y with $$\lambda=3$$.. Find the PDF $$f_W(w)$$, where $$W=X+Y$$.

I know that I simply use:

$$f_W(w) = \int\int (x+y) f_{X,Y}(x,y)dydx$$

The problem is that I don't know how to find their joint PDF. I know their PDF's separately:


$$f_X(x)=\left\{\begin{array}{cc}3e^{-3x},&<br /> x\geq 0\\0, & otherwise\end{array}\right.$$

$$f_Y(y)=\left\{\begin{array}{cc}4e^{-4x},&<br /> x\geq 0\\0, & otherwise\end{array}\right.$$

Would this help me in anyway? Please help.

 

[judoudo]

the joint density function is simply the product of the individual density functions
see here under independence:
http://en.wikipedia.org/wiki/Probability_density_function
in that article you also find the correct formula for the density of X+Y, what you have there seems to be the formula for E[X+Y] imho

 

[EugP]

the joint density function is simply the product of the individual density functions
see here under independence:
http://en.wikipedia.org/wiki/Probability_density_function
in that article you also find the correct formula for the density of X+Y, what you have there seems to be the formula for E[X+Y] imho

Yeah sorry I realized I made a mistake, and that link helped a lot. Thank you!