Page 5 - SUBHARADEV SEN AND AHMED

IAPQR Transactions

T 1 = min (X, Y),
T 2 = max (X, Y),

T 3 = X + Y. (5)
For any bivariate pdf, f(x, y), the densities of T 1, T 2 and T 3 are given by

, , , (6)

, , (7)

, , (8)

respectively.
In the BVFR situation, the densities take the following forms:

1 2 , 0 (9)

1 2 , 0 . (10)

,

1

4
, 2 2 , 2 ,

0 , (11)

where , 1 ; , 0, is the beta function.

If X and Y are independently distributed (i.e., α = 0), then the above three
densities reduce, to

, 0 ,

, 0 ,

,
, 0 .

Thus, the mean life or mean time to failure (MTF) for series system, with
pdf (9), comes out as

, say. (12)

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