Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 229If F X (x) takes the form given by Equation (7.93), we haveor1 exp‰ g…u n †Š ˆ 1exp‰g…u n †Šnˆ 1:1n ;…7:97†Now, consider F Y n(y) defined by Equation (7.89). In view of Equation (7.93),it takes the formF Yn …y† ˆf1 exp‰ g…y†Šg nexp‰g…u n †Š exp‰ g…y†Š nˆ 1n expf ‰g…y† g…u n †Šg nˆ 1:n…7:98†In the above, we have introduced into the equation the factor exp [g(u n )]/n,which is unity, as shown by Equation (7.97).Since u n is the mode or the ‘most likely’ value of Y n , function g(y) inEquation (7.98) can be expanded in powers of (y u n ) in the formg…y† ˆg…u n †‡ n …y u n †‡; …7:99†where n ˆ dg(y)/dy is evaluated at y ˆ u n . It is positive, as g(y) is an increasingfunction of y. Retaining only up to the linear term in Equation (7.99) andsubstituting it into Equation (7.98), we obtain exp‰ n …y u n †Š nF Yn …y† ˆ 1; …7:100†nin which n and u n are functions only of n and not of y. Using the identitylim 1 c nˆ exp… c†;n!1 nfor any real c, Equation (7.100) tends, as n !1, toF Y …y† ˆexpf exp‰ …y u†Šg; …7:101†which was to be proved. In arriving at Equation (7.101), we have assumed thatas n !1, F Y n(y) converges to F Y (y) as Y n converges to Y in some probabilisticsense.TLFeBOOK