Transform of Variable Using CDF Tech
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)
1) First, I think I solved this correctly as:
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?