If M Z n ( t ) = ( 3 e − t 4 n + e 3 t 4 n 4 ) 2 n Calculate lim n → ∞ M Z n ( t ) and interpret the result

Question

If      M                  Z                  n                      (  t  )  =            (                        3                      e                                                            −                  t                                                  4                                      n                                                                                +                      e                                                            3                  t                                                  4                                      n                                                                                      4            )              2      n      Calculate             lim              n        →        ∞                    M                        Z                      n                                (    t    )   and interpret the result

jmibanezla · Accepted Answer

This has the       1          ∞       form.So you can use this      lim          x      →      a        f  (  x      )          g      (      x      )        =  exp  ⁡  (      lim          x      →      a        (  f  (  x  )  −  1  )  g  (  x  )  )when       lim          x      →      a        f  (  x      )          g      (      x      )       is       1          ∞       form.Using the above we have :  exp  ⁡  (      lim          n      →      ∞                  3              e                                            −              t                                      4                              n                                                        +              e                                            3              t                                      4                              n                                                        −      4              4              2        n              )So inside the exponential we have       0    0   form and you can apply L&#039;Hospital.(just take   x  =      1    n   and then evaluate at   x  →      0    +   for easier calculation.)      lim          x      →      0                  3              e                                            −              t                              x                                      4                              +              e                                            3              t                              x                                      4                              −      4              2      x      Using   x  =      h          2      .(Continuous function) We have:-      lim          h      →      0                  3              e                                            −              t              h                        4                              +              e                                            3              t              h                        4                              −      4              2              h                  2                    Again this is       0    0   form . So again L&#039;Hospitaling it      lim          h      →      0                                    3                      t                          2                                      16                    e                                            −              t              h                        4                              +                        9                      t                          2                                      16                    e                                            3              t              h                        4                                4    =            12              t                  2                            16      ⋅      4        =            3              t                  2                      16  So the original limit converges to  exp  ⁡  (            3              t                  2                      16    )This just means that the MGF of the sequence of Rv&#039;s converges to the MGF of an   N  (  0  ,      3    8    ) distribution.Where   N  (  μ  ,      σ          2        ) denotes the normal distribution with mean   μ and standard deviation   σ

If M <mrow class="MJX-TeXAtom-ORD"> Z <mrow class="MJX-TeXAtom-O

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New Questions in Calculus 1