-
30. Central limit theorem
In the study of probability theory, the central limit theorem (CLT) states that the distribution of sample means approximates a normal distribution, as the sample size becomes larger, assuming that all samples are identical in size, and regardless of the population distribution shape. Said another way, CLT is a statistical...
-
29. Chebyshev’s inequality and the proof of the weak law of large numbers
Chebyshev’s inequality Chebyshev’s inequality guarantees that, for a wide class of probability distributions, no more than a certain fraction of values can be more than a certain distance from the mean. Specifically, no more than 1/k^2 of the distribution’s values can be more than k standard deviations away from the...
-
28. Do a research about the Box Muller Transform.
A Box Muller transform takes a continuous, two dimensional uniform distribution and transforms it to a normal distribution. It is widely used in statistical sampling, and is an easy to run, elegant way to come up with a standard normal model. In fact, since it can be used to generate...
-
27. Find and implement the most common algorithms for Normal Random Variables generation.
Some existing methods for generating standard normal random numbers discussed in this section. 1. Sum of Uniform Random Variables The simplest way of generating normal variables is an application of the central limit theorem. The central limit theorem is a weak convergence result that expresses the fact that any sum...
-
26. Do a research about the inversion method to generate random variables.
Inverse transform sampling is a method for generating random numbers from any probability distribution by using its inverse cumulative distribution F−1(x). Recall that the cumulative distribution for a random variable X is FX(x)=P(X≤x). In what follows, we assume that our computer can, on demand, generate independent realizations of a random...
-
25. Do a research about the most interesting statistical charts and make plans to include them in your own personal library.
One goal of statistics is to present data in a meaningful way. Often, data sets involve millions (if not billions) of values. This is far too many to print out in a journal article or sidebar of a magazine story. That’s where graphs can be invaluable, allowing statisticians to provide...
-
24. Interpretation and importance of the Bayes Theorem in statistical inference and the concept of prior, posterior probabilities and the likelihood.
Bayes’ theorem is a formula that describes how to update the probabilities of hypotheses when given evidence. It follows simply from the axioms of conditional probability, but can be used to powerfully reason about a wide range of problems involving belief updates. It allows you to update predicted probabilities of...
-
23. The concept of sampling distribution and particularly the sampling distribution of the 𝞵 and the 𝜎.
Statistical sampling is used quite often in statistics. In this process, we aim to determine something about a population. Since populations are typically large in size, we form a statistical sample by selecting a subset of the population that is of a predetermined size. By studying the sample we can...
