Parent: Probability theory

Expectation

Source: Bishop

Definition

$$ \begin{alignat}{3} \mathbb{E}[X] &= \sum_{x \in X} x \cdot P(x) &&\qquad \text{discrete}\
&= \int_{X} x \cdot p(x) ~ \text{d}x &&\qquad \text{continuous} \end{alignat} $$

Approximation

Given a finite number of $N$ points drawn from the probability distribution, $\mathbb{E}$ can be approximated by $$ \mathbb{E}[X] \approx \dfrac{1}{N} \sum_{n=1}^N X(\omega_n) $$

Source:  rlabbe

Example

• If we take a thousand sensor readings, the readings won’t always be the same (due to the inherent noise).
• The expected value ‘averages’ all of the readings into a single value.
• This can be a mean (probabilities of all values assumed equal)
• Or if incorporating individual and different probabilities, the expectation isn’t the mean of the range of values

$$ \begin{alignat}{3} \mathbb{E}[X] &= \sum_{i=1}^n p_i x_i &&\qquad \text{discrete}\
&= \int_{a}^b x f(x) ~ \text{d}x &&\qquad \text{continuous} \end{alignat} $$

Proven: the average of a large number of measurements will be very close to the actual weight

• What is the name for this theorem?

Summary

• Using a normal mean calculation assumes that all measurements are equally likely (assumption of equal probability).
• However, real sensors tend to return readings close to the actual value (barring any offset disturbances).
• They can still return readings further away from the actual value, just with a reduced probability compared to values nearer the actual value
• The probability distribution must be factored in somehow