# Read e-book Elementary Inequalities - Tutorial Text No. 1

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Example 2: For an example where Markov inequality gives a bad result, let us the example of a dice. Let X be the face that shows up when we toss it. By Markov inequality,. The upper bound is greater than 1! Of course using axioms of probability, we can set it to 1 while the actual probability is closer to 0. You can play around with the coin example or the score example to find cases where Markov inequality provides really weak results.

The last example might have made you think that the Markov inequality is useless. On the contrary, it provided a weak bound because the amount of information we provided to it is limited.

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All we provided to it were that the variable is non negative and that the expected value is known and finite. In this section, we will show that it is indeed tight — that is Markov inequality is already doing as much as it can. From the previous example, we can see an example where Markov inequality is tight. If the mean of students is 20 and if 50 students got a score of exactly 0, then Markov implies that atmost 50 students can get a score of atleast We can see that ,.

This implies that the bound is actually tight! Of course one of the reasons why it was tight is that the other value is 0 and the value of the random variable is exactly k. This is consistent with the score example we saw above. Chebyshev inequality is another powerful tool that we can use. In this inequality, we remove the restriction that the random variable has to be non negative. As a price, we now need to know additional information about the variable — finite expected value and finite variance. In contrast to Markov, Chebyshev allows you to estimate the deviation of the random variable from its mean.

A common use of it estimates the probability of the deviation from its mean in terms of its standard deviation. Similar to Markov inequality, we can state two variants of Chebyshev. Let us first take a look at the simplest version. Given a random variable X and its finite mean and variance, we can bound the deviation as.

There are few interesting things to observe here : 1 In contrast to Markov inequality, Chebyshev inequality allows you to bound the deviation on both sides of the mean. A more general Chebyshev inequality bounds the deviation from mean to any constant a.

### Obtaining Gmsh

Given a positive constant a ,. The proof of this inequality is straightforward and comes from a clever application of Markov inequality. As discussed above we select. Using it we get ,. We used the Markov inequality in the second line and used the fact that. It is important to notice that Chebyshev provides bound on both sides of the error.

One common mistake to do when applying Chebyshev is to divide the resulting probabilistic bound by 2 to get one sided error.

## What Is Markov's Inequality?

This is valid only if the distribution is symmetric. Else it will give incorrect results. You can refer Wikipedia to see one sided Chebyshev inequalities. One of the neat applications of Chebyshev inequality is to use it for higher moments. As you would have observed, in Markov inequality, we used only the first moment. In the Chebyshev inequality, we use the second moment and first. We can use the proof above to adapt Chebyshev inequality for higher moments. In this post, I will give a simple argument for even moments only. For general argument odd and even look at this Math Overflow post.

The proof of Chebyshev for higher moments is almost exactly the same as the one above. The only observation we make is that is always non negative for any k. It should be intuitive to note that the more information we get the tighter the bound is. Using Chebyshev inequality, we previously claimed that atmost one fourth of the values that X can take is beyond 2 standard deviation of the mean.

It is possible to turn this statement around to get a confidence interval. More generally, we can claim that, percentage of the population lies in the interval. We previously saw two applications of Chebyshev inequality — One to get tighter bounds using higher moments without using complex inequalities. The other is to estimate confidence interval.

There are some other cool applications that we will state without providing the proof. For proofs refer the Wikipedia entry on Chebyshev inequality. Similar to Markov inequality, we can prove the tightness of Chebyshev inequality. I had fun deriving this proof and hopefully some one will find it useful. Define a random variable X as ,. If we want to find the probability that the variable deviates from mean by constant C, the bound provided by Chebyshev is ,. Markov and Chebyshev inequalities are two of the simplest , yet very powerful inequalities.

Clever application of them provide very useful bounds without knowing anything about the distribution of the random variable. Markov inequality bounds the probability that a nonnegative random variable exceeds any multiple of its expected value or any constant. Chebyshev does not expect the variable to non negative but needs additional information to provide a tighter bound. Both Markov and Chebyshev inequalities are tight — This means with the information provided, the inequalities provide the most information they can provide.

Posted in math Tagged chebyshev , inequalities , markov , probability 18 Comments. Lovely write up. I was reading up on the net to find out whether one can pictorially represent the Expectation and the Chebychev Inequality. I found your blog and read it with interest. Please continue the erudite work.

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Thanks for the kind words! We will also define the Wronskian and show how it can be used to determine if a pair of solutions are a fundamental set of solutions. More on the Wronskian — In this section we will examine how the Wronskian, introduced in the previous section, can be used to determine if two functions are linearly independent or linearly dependent. We will also give and an alternate method for finding the Wronskian. Nonhomogeneous Differential Equations — In this section we will discuss the basics of solving nonhomogeneous differential equations. We define the complimentary and particular solution and give the form of the general solution to a nonhomogeneous differential equation.

Undetermined Coefficients — In this section we introduce the method of undetermined coefficients to find particular solutions to nonhomogeneous differential equation. We work a wide variety of examples illustrating the many guidelines for making the initial guess of the form of the particular solution that is needed for the method. Variation of Parameters — In this section we introduce the method of variation of parameters to find particular solutions to nonhomogeneous differential equation. We give a detailed examination of the method as well as derive a formula that can be used to find particular solutions.

Mechanical Vibrations — In this section we will examine mechanical vibrations. In particular we will model an object connected to a spring and moving up and down. We also allow for the introduction of a damper to the system and for general external forces to act on the object. Note as well that while we example mechanical vibrations in this section a simple change of notation and corresponding change in what the quantities represent can move this into almost any other engineering field.

The Definition — In this section we give the definition of the Laplace transform. We will also compute a couple Laplace transforms using the definition. Laplace Transforms — In this section we introduce the way we usually compute Laplace transforms that avoids needing to use the definition. We discuss the table of Laplace transforms used in this material and work a variety of examples illustrating the use of the table of Laplace transforms.

Inverse Laplace Transforms — In this section we ask the opposite question from the previous section. In other words, given a Laplace transform, what function did we originally have?

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4. We again work a variety of examples illustrating how to use the table of Laplace transforms to do this as well as some of the manipulation of the given Laplace transform that is needed in order to use the table. Step Functions — In this section we introduce the step or Heaviside function. We illustrate how to write a piecewise function in terms of Heaviside functions.

We also work a variety of examples showing how to take Laplace transforms and inverse Laplace transforms that involve Heaviside functions. We also derive the formulas for taking the Laplace transform of functions which involve Heaviside functions. The examples in this section are restricted to differential equations that could be solved without using Laplace transform.

The advantage of starting out with this type of differential equation is that the work tends to be not as involved and we can always check our answers if we wish to. We do not work a great many examples in this section. We only work a couple to illustrate how the process works with Laplace transforms. Without Laplace transforms solving these would involve quite a bit of work. While we do work one of these examples without Laplace transforms, we do it only to show what would be involved if we did try to solve one of the examples without using Laplace transforms.

We work a couple of examples of solving differential equations involving Dirac Delta functions and unlike problems with Heaviside functions our only real option for this kind of differential equation is to use Laplace transforms.

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