For reference, I have added theorem Theorem 26 (Theorem of de Moivre - Laplace) Let X1, X2, ... be independent random variables with Bernoulli(p)distribution. LetS, = > X;.ThenSn -пр< a= ¤(a),lim Pfor every a E R.(Vmp(1 – p)51In particular, when n is large (as a rule of thumb, when n > 30), then the cdf of the rescaled binomial random variableSn is approximately equal to the cdf of a standard-normal random variable ZN(0, 1), i.e.Snпр< a~ P(Z < a) = ¢(a).Упр(1 — р) Exercise 50(i) Let Xn ~ Poisson(10), n > 1, be independent random variables. Estimate the probability thatP(9 < S40 < 12).(ii) Let XnExp(4), n 2 1, be independent random variables. Estimate the probability that1P(T0 S100 2

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Description: For reference, I have added theorem Theorem 26 (Theorem of de Moivre - Laplace) Let X1, X2, ... be independent random variables with Bernoulli(p)distribution. LetS, = > X;.ThenSn -пр< a= ¤(a),lim Pfor every a E R.(Vmp(1 – p)51In particular, when n is

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Exercise 50 (i) Let Xn Poisson(10), n > 1, be independent random variables. Estimate the probability that 2- P(9 < S40 < 12).

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For reference, I have added theorem

Theorem 26 (Theorem of de Moivre - Laplace) Let X1, X2, ... be independent random variables with Bernoulli(p)
distribution. Let
S, = > X;.
Then
Sn -
пр
< a
= ¤(a),
lim P
for every a E R.
(Vmp(1 – p)
51
In particular, when n is large (as a rule of thumb, when n > 30), then the cdf of the rescaled binomial random variable
Sn is approximately equal to the cdf of a standard-normal random variable Z
N(0, 1), i.e.
Sn
пр
< a
~ P(Z < a) = ¢(a).
Упр(1 — р)

Exercise 50
(i) Let Xn ~ Poisson(10), n > 1, be independent random variables. Estimate the probability that
P(9 < S40 < 12).
(ii) Let Xn
Exp(4), n 2 1, be independent random variables. Estimate the probability that
1
P(T0 S100 2

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