Probability Theory Mastering

In the previous chapters we considered how it is possible to estimate the parameters of the population and check the quality of the data of the estimates. But those estimates were point: we simply determined the possible value of the parameter based on the data we have. But there is another approach: we can construct a certain interval that, with some probability, covers the real value of the desired parameter. This interval is called the confidence interval. Let's look at the definition:

The principle of constructing confidence intervals is somewhat similar to the principle of constructing point estimates. We also use a certain function with our samples as arguments for this function. That we use the distribution law of this function and build an interval. But a rigorous mathematical explanation of this process can be quite complicated, so we will not stop on it in more detail.

Note

It is important to admit that there is also another type of interval estimation of population parameters - credible interval, built using the Bayesian theorem. These intervals have different interpretations; the confidence interval is essentially an interval with random ends that, with a certain probability, covers the real constant value of the parameter; at the same time, the credible interval is a constant interval in which the random value of the desired parameter falls with a certain probability.

Let's look at how to build a confidence interval for Gaussian distribution expectation parameter. We will consider 2 different situations:

In the image above, we provided a confidence interval for Gaussian expectation if we know variance. We use the PPF of Gaussian distribution and sample to build this interval.

Then we provided a confidence interval for Gaussian expectation if we don't know the variance and used adjusted sample variance instead of known variance for estimation. We use the PPF of Student distribution with n-1 degree of freedom to build this interval.
Let's now look at how to build a confidence interval for the mean value of Gaussian samples in Python. We will use different confidence levels and compare intervals built due to corresponding confidence level.

We see that the higher the confidence level, the wider the interval we get. This is quite logical, since the wider the interval, the higher the probability that this interval covers the real value of the mean.

What is confidence level of confidence interval?

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Probability that confidence interval will cover corresponding parameter

Probability that confidence interval will not cover corresponding parameter

Probability that parameter will fall into the corresponding interval

Probability that parameter will not fall into the corresponding interval

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