Random Variable

Definition

It is a variable whose value is dependent on the outcome of a random experiment. It is defined as a mapping from the sample space to the real numbers.

• Continuous Random Variable: If a Random Variable can take an Infinite number of values within a range or interval. Then it is is a Continuous random variable.
• Discrete Random Variable: Random Variables whose sample set is a fixed and is not a range.

Note: Random variables are denoted by uppercase letters.

Example

• Player is tolling a die, a win or loss is associated with each outcome

$$\begin{array}{ccccccc}\theta & 1& 2& 3& 4& 5& 6\\ \text{win/loss}& 1& 2& 3& -4& -5& -6\end{array}$$

• The game can be judged based on the expected value

$$\mathbb{E}[X]=\sum _{i=1}^{6}X\left({\theta }_i\right)\frac{1}{6}=-1.5$$

Note: Here $X$ is the random variable.