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Random Variables, Expectation and Variances

Random Variables

The random variables are generally defined as the variable whose value is generally unknown or it can be defined as the function which takes the value of the experiment outcomes.

Types of the Random Variables

  1. Discrete Random Variable
  2. Random Random Variable

Expectation
The expectation is nothing but it is a long time average value of the experiment. It can also be called as the mean or expected value and it can be represented as E(X) or µ(X).
If X is the continues random variable with the probability density function fx(x). The expectation formula is given by,

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In case of the discrete random variable, the expected value can be calculated by the below formula,

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Properties of Expectation
For any random variable X that is discrete or continuous,
E (aX+b) = aE (x)+b

For any two random variables X and Y, that is discrete, continuous, dependent or non-independent,
E(X+Y) = E(X) + E(Y)

For the expectation, the general multiplication rule is only possible if only the two variables are independent.
E (XY) = E(X)E(Y)

Variance
The variance of the random variables is generally used to find out how far the values are generally away from the mean.
The variance of the  X is given by,
Var (X) = E(X2) – ( E(X) )2
If the random variable X has high variance then the values are spread a long way from the mean and vice versa.
Example,
Let X be one of the continuous random variable with probability density function,
FX(x) = { 2x-2  for 1<x<2 , 0 for otherwise }
Calculate the mean and the variance.

Mean
Using the formula from the above the mean value can be calculated as,

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        Similarly, the variance can be calculated using the below formula,

Var ( X ) = E( X2) – (E(X))2
             = 2 – {2 log(2)}2
              = 0.0782

Reference

https://www.stat.auckland.ac.nz/~fewster/325/notes/ch3.pdf

http://theanalysisofdata.com/probability/2_3.html

https://revisionmaths.com/advanced-level-maths-revision/statistics/expectation-and-variance

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