Fundamentals of Probability and Statistics for Engineers

which is the desired solution. If random variables X 1 and X 2 are independent,
straightforward differentiation shows that

Let us note here that the results given above can be obtained following a
different, and more direct, procedure. Note that the event [min (X 1 ,X 2
equivalent to the event (X 1 2 y). Hence,

Since

we have

If X 1 and X 2 are independent, we have

and

x 2

x 1

y

y

R^2

Figure 5. 19 Region R^2 in Example 5.15

144 Fundamentals of Probability and Statistics for Engineers

y X

fY…y†ˆfX 1 …y†‰ 1

 FX 2 …y†Š‡fX 2 …y†‰ 1

 FX 1 …y†Š:

 [ 

FY…y†ˆP…Yy†ˆP‰min…X 1 ;X 2 †yŠ

ˆP…X 1 y[X 2 y†:

P…A[B†ˆP…A†‡P…B†

P…AB†;

FY…y†ˆP…X 1 y†‡P…X 2 y†

P…X 1 y\X 2 y†

ˆFX 1 …y†‡FX 2 …y†

FX 1 X 2 …y;y†:

FY…y†ˆFX 1 …y†‡FX 2 …y†

FX 1 …y†FX 2 …y†;

fY…y†ˆ

dFY…y†

dy

ˆfX 1 …y†‰ 1

 FX 2 …y†Š‡fX 2 …y†‰ 1

 FX 1 …y†Š:

) y]is