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Which inequality is based on expectation only?
Chebyshev’s Inequality. Markov’s inequality gives us upper bounds on the tail probabilities of a non-negative random variable, based only on the expectation.
Is Markov inequality tight?
Even though Markov’s and Chebyshev’s Inequality only use information about the expectation and thevariance of the random variable under consideration, they are essentially tight for a general random variable.
Why do we use Chebyshev’s inequality?
The rule is often called Chebyshev’s theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.
Which is an example of an inequality involving expectations?
Both involve expectations (and the theorems derived in the previous chapter). Jensen’s Inequality is a statement about the relative size of the expectation of a function compared with the function over that expectation (with respect to some random variable).
Jensen’s Inequality is a statement about the relative size of the expectation of a function compared with the function over that expectation (with respect to some random variable). To understand the mechanics, I first define convex functions and then walkthrough the logic behind the inequality itself.
Why is the first inequality of the integrand true?
I can see why the first inequality is true. Basically, it is rephrase of the triangular inequality, i.e. the fact that the absolute value of an integral is less than or equal to the integral of the absolute value of the integrand.
How is the Cauchy-Schwarz inequality used in mathematics?
Cauchy-Schwarz inequality has been applied to various subjects such as probability theory. I wonder how to prove the following version of the Cauchy-Schwarz inequality for random variables: