Exercise 49 Let X ~ NegBinom(900, 4). Estimate the probability P(X > 3000) (i) with the Markov inequality, (ii) with the normal distribution. Hint: Use the fact that a negative binomial random variable can be written as a sum of geometric random variables. Exercise 50 (i) Let X, ~ Poisson(10), n > 1, be independent random variables. Estimate the probability that P(9 < S40 < 12). (ii) Let X, ~ Exp(4), n > 1, be independent random variables. Estimate the probability that P S100 2 100
Exercise 49 Let X ~ NegBinom(900, 4). Estimate the probability P(X > 3000) (i) with the Markov inequality, (ii) with the normal distribution. Hint: Use the fact that a negative binomial random variable can be written as a sum of geometric random variables. Exercise 50 (i) Let X, ~ Poisson(10), n > 1, be independent random variables. Estimate the probability that P(9 < S40 < 12). (ii) Let X, ~ Exp(4), n > 1, be independent random variables. Estimate the probability that P S100 2 100
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
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I put the exercise before but I see the comment that the question is not complete. So I need exercise 49 and 50
Transcribed Image Text:Theorem 27 (CENTRAL LIMIT THEOREM (CLT)) Let X1, X2, ... be independent identically distributed random vari-
ables with finite expectation and finite variance. Let
Sn
ΣΧ.
i=1
Then
S, – ES,
<a) = ¢(a),
for every a E R.
lim P
Var(S„)
n00
In particular, when n is large (as a rule of thumb, when n > 30), then the cdf of the rescaled random variable S, is
approximately equal to the cdf of a standard-normal random variable Z - N (0,1), i.e.
S, - ES,
P
<a - P(Z < a) = ¢(a).
Var(S,)
Exercise 49 Let X•
NegBinom(900, 4). Estimate the probability
P(X 2 3000)
(i) with the Markov inequality,
(ii) with the normal distribution.
Hint: Use the fact that a negative binomial random variable can be written as a sum of geometric random variables.
Exercise 50 (i) Let X„ ~ Poisson(10), n > 1, be independent random variables. Estimate the probability that
P(9 < S40 < 12).
(ii) Let X„ ~ Exp(4), n 2 1, be independent random variables. Estimate the probability that
P
S100 2
100
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