Answer to Question #129398 in Statistics and Probability for Usama

Question

Explain the concept of a random variable. Define a Continuous random variable and what
are its properties?

Answer 1

Random variable is a variable which, as a result of the test, will obtain a value depending on random factors and unpredictable in advance. Random variables can be divided into discrete and continuous.

Example 1. We are tossing a coin. It has head and tail and we cannot predict what side we will see in every experiment. The side of the coin is random variable. Our sample space is $\Omega$ = {head, tail}. The variable is called discrete if the $\Omega$ is finite or countable. We can introduce such a function that will map $\Omega(\omega_1, \omega_2, ...): \omega \rightarrow p_i$ where $p_i$ is a probability of particular possible result of experiment from the sample set.

Example 2. We measure the precise current in some electronic element and see that we can not predict the value of current (because of random noises due to quantum effects). In this case value of current can be in principle anything and can be changed continuously, so this is continuous random variable. $\Omega$ = {..., -5.32, ..., -1.258, .., 0, 5.657, ...}. This sample space is continuous. The properties of continuous random variable are similar to discrete random variables. The only difference is that all sums (as, for example, in mean or variance) convert to integrals (in some sense it was expected because integral is a sum in continuous space).

For illustration:

$\bar{x} = \frac{\sum_{i=1}^N x_i}{N} \Rightarrow \bar{x} = \smallint_\Omega xdF(x)$

where F(x) shows how random variable is distributed.