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

Adriana Ayers 2022-06-27 Answered
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
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Answers (1)

jmibanezla
Answered 2022-06-28 Author has 17 answers
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'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'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'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

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