Expected Value

Definition

It is the average value (mean) of a random experiment. 

Expected Value for

1. **Discrete Random variable **

   $$\mathbb{E}[X]:=\sum _{i=1}^{n}X({\theta }_i)\cdot P(X({\theta }_i))$$

2. Continuous Random variable

   $$\begin{array}{rl}\mathbb{E}[X]& :={\int }_{\Theta }X(\theta )d\mathbb{P}(\theta )\\ & \text{or}\\ \mathbb{E}[X]& :={\int }_{-\infty }^{\infty }x{f}_X(x)\mathrm{d}x\end{array}$$

where

• $n\to$ size of sample space
• $\theta \to$ sample space
• $X\to$random variable
• $x\to$ possible value of random variable $x=X(\theta )$
• $P\to$ Probability
• $f_X\to$ PDF

Linearity of Expectation

$$\mathbb{E}[A+B]=\mathbb{E}[A]+\mathbb{E}[B]$$

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References

1. https://brilliant.org/wiki/expected-value/
2. Uncertainty Analysis Script Pg.12