Transform of Variable Using CDF Tech

Problem:

Let x1, x2 be independent random variables representing lifetimes (in hours) of two key components of a device, which fails when and only when both components fail. Say Xi has the exponential distribution with mean 1000. Let Y1 = min(X1,X2) and Y2 = max(X1,X2); so the space of Y1,Y2 is 0<y1<y2< inf+

Find G(y1,y2) = P(Y1<=y1, Y2<=y2)

Steps Taken:

1) First, I think I solved this correctly as:

P(Y1<=y1, Y2<=y2)

=

P(Y1<=y1)* P(Y2<=y2)

=

[1-[1-P(X1<=y1)[1-P(X2<=y1)][P(X1<=y2)P(X2<=y2)]

but the book gives a much different form of an answer.

2) I multiply two exponential distributions together to get

1/1000000 * exp( (-x1-x2)/1000). This is the joint pdf.

3) I wanted to draw a picture of the first quadrant (Y1 on the horizontal and Y2 on the vertical) and shade everything in Q1 above the line y2=y1.

Then I'd integrate the joint pdf wrt Y2 first and then Y1, with the limits y1 to y2 and 0 to y1 respectively to get the asked for cdf.

The book gives an answer with a 2 multiplied by the joint pdf I dont have and I cant figure out why (i.e I am missing the important logic). Anyone help?

Thanks!