Unizor - Statistics - Volume of Data

Unizor - Creative Minds through Art of Mathematics - Math4Teens

Notes to a video lecture on http://www.unizor.com

Volume of Statistical Data

As we noted, the purpose of Mathematical Statistics is, using the past observations, to evaluate the probability of certain events to be able to predict their occurrence in the future.

Assume, we are dealing with a discrete random variable ξ that can take any value from a set
{X1, X2...XN}
with corresponding probabilities
{P1, P2...PN},
but neither values nor probabilities are known to us.

To determine these values and probabilities, that is to determine the distribution of probabilities of random variable ξ, we've made certain number of random experiments (making sure, the conditions of experiments are the same and independent from each other) and observed the values our random variable took.

Let's state up front that to determine these probabilistic characteristics of our random variable precisely in finite amount of time is impossible.

In the Theory of Probabilities part of this course we have defined the probability Pi of a discrete random variable ξ to take specific value Xi as a limit of its frequency of taking this value as the number of experiments tends to infinity.

Therefore, measuring this frequency during some finite time, making only finite number of experiments, is possible only approximately. However, we might think that, as the number of experiments increases, the precision of our approximation is getting better and better.

In most cases this is true, also, strictly speaking, we have to define quantitatively the quality of approximation and agree when to call it sufficient for our purposes.