B3 Consider X ~ Bern(p) (a) Find Mx(t), the moment generating function of X. iid (b) If X1,..., Xn Bern(p), find the MGF, say My (t) of n Y = ΣΧ (c) Using the fact that i=1 n lim (1 (1+2)"= N→X = e² find limn→∞ My (t) in the case that p satisfies limn→∞ np = λ, say. (d) State the distribution of Y in the case that n is not large, and the distribution of Y in the limiting case described in the question.
B3 Consider X ~ Bern(p) (a) Find Mx(t), the moment generating function of X. iid (b) If X1,..., Xn Bern(p), find the MGF, say My (t) of n Y = ΣΧ (c) Using the fact that i=1 n lim (1 (1+2)"= N→X = e² find limn→∞ My (t) in the case that p satisfies limn→∞ np = λ, say. (d) State the distribution of Y in the case that n is not large, and the distribution of Y in the limiting case described in the question.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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Transcribed Image Text:B3 Consider X ~ Bern(p)
(a) Find Mx(t), the moment generating function of X.
iid
(b) If X1,..., Xn
Bern(p), find the MGF, say My (t) of
n
Y =
ΣΧ
(c) Using the fact that
i=1
n
lim (1
(1+2)"=
N→X
= e²
find limn→∞ My (t) in the case that p satisfies limn→∞ np = λ, say.
(d) State the distribution of Y in the case that n is not large, and the distribution of Y in
the limiting case described in the question.
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