i) Show that (Hint: Use the CLT for Xn = Xį ~ Poi(1)). lim e n→∞ -n k! 1 2 k=0 1 Xi, where X₁ are i.i.d. random variables such that

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Please if able provide some explanation with the taken steps,thank you in advance.

i)
Show that
(Hint: Use the CLT for Xn
Xi ~ Poi(1)).
ii)(
=
lim e
n→∞
-n
n
k!
1
2
k=0
1 X₁, where Xį are i.i.d. random variables such that
Let X₁,..., Xn be i.i.d independent Poisson random variables each having
mean 1 (i.e., Xį ~ Poi(1)). Let Sn := Σ₁ X₁. Using the above information, try to prove the
Stirling approximation:
n! ≈ √√√2πn²+¹/2 e-n
Transcribed Image Text:i) Show that (Hint: Use the CLT for Xn Xi ~ Poi(1)). ii)( = lim e n→∞ -n n k! 1 2 k=0 1 X₁, where Xį are i.i.d. random variables such that Let X₁,..., Xn be i.i.d independent Poisson random variables each having mean 1 (i.e., Xį ~ Poi(1)). Let Sn := Σ₁ X₁. Using the above information, try to prove the Stirling approximation: n! ≈ √√√2πn²+¹/2 e-n
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