Probability that X fails before Y with exponential distributions

[tony3333]

hi,
i have a problem and i really want you to help me with it.
if we have: density of X : f(x)=exp(-x), and density of Y : f(y)=2*exp(-2y) (independent components), what is the probability that X component fails first?
(it should be a number)
thank you

 

[micromass]

If I understand correctly, you will want P{X<Y}.

You can calculate this as follows: let $$A=\{(x,y)~\vert~0\leq x\leq y\}$$
Then you need to calculate

$$\int\int_A{2e^{-x}e^{-2y}dxdy}$$

this will be your probability...

 

[tony3333]

If I understand correctly, you will want P{X<Y}.

You can calculate this as follows: let $$A=\{(x,y)~\vert~0\leq x\leq y\}$$
Then you need to calculate

$$\int\int_A{2e^{-x}e^{-2y}dxdy}$$

this will be your probability...

and because these are independent, we can separate the integrals and have

$$\int_A{e^{-x}dx \int_A{2e^{-2y}dy$$

and then the variables of every probability , x,y, can be descibed by the same symbol, let's say t
but what should be the limits of the integral?