The equation mentioned by Amar Nath Yadav defines so called "intensity averaged lifetime" and is valid only under specific circumstances. There is also another equation, so called "amplitude averaged lifetime", defined in this paper: (1) Article The Correct Use of “Average” Fluorescence Parameters
The choice of appropriate formula for average lifetime is not that obvious. Recently, we wrote a paper that explains the differences between two different formulas for average lifetime: (2)
Article On the choice of proper average lifetime formula for an ense...
In this paper, we use equations equivalent to formulas given in (1) but involving simple integrals (this has the advantage that no fitting is required). Obviously, everything stated above is important only in the case of non-single exponential decays.