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Dispersion, Confidence Intervals, and More!

In order to find out how correct the obtained results are, let’s go a little bit deeper into math1ematical statistics.

As we have already mentioned in the beginning of this article, many random phenomena, including the natural ones, obey to the normal Gaussian distribution law. Even during the HDD tests we can come across this law. This is what the distribution of HDD random access time looks like:

Therefore, let’s assume that the HDD performance we measure equals [Actual HDD Speed + Error], where the actual HDD speed is a constant, and the error is random. Then the distribution of random HDD performance values will be normal for a definitely higher number of measurements.

The most important measuring distribution parameters are math1ematical expectation M(x) and dispersion D(x) of the random variable x. The distribution parameters of the random variable x are usually unknown in the math1ematical statistics tasks. The researcher usually has at his disposal only a sample of independent experiments of the size n [x1, x2, …, xn]. In this case the sampling parameters are derived from the sample and then serve as a certain approximation of the theoretical or general parameters. The larger is the sample n, the better is the approximation. In practice we can consider sampling parameters as coinciding with general parameters in case of n>50.

Let’s take a closer look at these parameters and their features.

The math1ematical expectation of the continuous random variable is set by the following integral:

For the discrete random variable the formula looks as follows:

Where x(i) and p(i) are separate values and corresponding probabilities of the random variables, and n – the number of its possible values.

In a particular case for the even distribution of a random variable with n possible values we get:

In other words, the math1ematical expectation coincides with the notion of the arithmetic mean value. In the general case when both events are not equally probable

the math1ematical expectation equals the so called average value of the discrete random variable, when the different probabilities of the individual values are taken into account.

To cut the long story short, math1ematical expectation is a value with the random variable values surrounding it.

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