Exercise 48 If we throw a fair coin 10,000 times, what is the probability that it lands on “tail” at least 5,500 times? (i) Write down the exact probability in sum notation. (ii) Estimate the probability with the Chebyshev inequality. (iii) Estimate the probability with the Poisson‐Approximation. (iv) Estimate die probability with the normal distribution. Exercise 49 Let X ∼ NegBinom(900, 1/4 ). Estimate the probability P(X ≥ 3000)
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
Exercise 48 If we throw a fair coin 10,000 times, what is the
(i) Write down the exact probability in sum notation.
(ii) Estimate the probability with the Chebyshev inequality.
(iii) Estimate the probability with the Poisson‐Approximation.
(iv) Estimate die probability with the
Exercise 49 Let X ∼ NegBinom(900, 1/4 ). 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 Xn ∼ Poisson(10), n ≥ 1, be independent random variables. Estimate the probability that P(9 ≤ S40 ≤ 12).
(ii) Let Xn ∼ Exp(4), n ≥ 1, be independent random variables. Estimate the probability that P (1/100 * S100 ≥ 1 2).
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