Suppose X and Y have joint density f(x,y)=e^(-(x+y)) for x,y0. Find the distribution function.

Question

Suppose X and Y have joint density f(x,y)=e^(-(x+y)) for x,y>0.  Find the distribution function.

anonymous · Answer

Which distribution function?

anonymous · Answer

It looks like X and Y are two independent exponential distribution functions.

anonymous · Answer

And both have a mean of $1$.

anonymous · Answer

Exponential function is given by:$$
\lambda e^{-\lambda x}
$$

anonymous · Answer

Where the mean is $\lambda ^{-1}$

turingtest · Answer

@haycamille are you looking for the cumulative distribution function $F_{X,Y}(x,y)=\iint f_{X,Y}(x,y)dxdy$ ?

turingtest · Answer

integrals from -infty to x and y...

anonymous · Answer

Yes i think so. All the book tells me to do is find the distribution function, so I would assume that's what I'm looking for.

turingtest · Answer

well that's the only "distribution" function I know of; pdf's are "density" functions.
simply apply the definition of the CDF (cumulative density function) for a joint pdf$$F_{X,Y}(x,y)=\int_{-\infty}^x\int_{-\infty}^yf_{X,Y}(s,t)dsdt$$

turingtest · Answer

s and t are just dummy variables

anonymous · Answer

So since x and y are both greater than 0, my bounds for the integral would be from 0 to infinity for both x and y?

turingtest · Answer

cumulative distribution function*
I keep mixing those up :P
and "yes" to your question

anonymous · Answer

Okay thank you so much!

turingtest · Answer

welcome :)