What Bernoulli inequality is
The Bernoulli inequality is an inequality which states that for any real number x greater than or equal to -1, the following holds: (1+x)^n ≥ 1 + nx, where n is any positive integer.
The Bernoulli inequality is a useful tool in proving other inequalities, particularly when dealing with summations. It is a generalization of the binomial theorem, which states that (1+x)^n = ΣnCkx^k, where nCk is the binomial coefficient.
The Bernoulli inequality can be proved using induction. The base case for the induction is n = 1, for which the inequality holds since (1+x)^1 = 1+x ≥ 1+x.
Now for the induction step, assume the inequality holds for n = k. Then, the left side of the inequality can be rewritten as (1+x)^k+1 = (1+x)^k(1+x) ≥ (1+kx)(1+x) = 1+(k+1)x. The right side of the inequality is 1+(k+1)x, so the left side is greater than or equal to the right side, as required. Thus, the inequality holds for n = k+1.
By induction, the Bernoulli inequality holds for any positive integer n.
Examples
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Bernoulli inequality is used to bound the probability of success in a binomial random variable. For example, if the probability of success in a coin toss is 0.5, the Bernoulli inequality can be used to bound the probability of obtaining at least 3 heads in 4 tosses.
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Bernoulli inequality is useful for proving inequalities in probability theory, such as Chebyshev’s inequality. For example, the Bernoulli inequality can be used to prove that the probability of observing a result more than two standard deviations away from the mean is at most 0.05.
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Bernoulli inequality is also used to prove inequalities in statistics, such as the law of large numbers. For example, the Bernoulli inequality can be used to prove that the average of a large number of independent random variables approaches its expected value as the number of trials increases.