27 lines
757 B
Markdown
27 lines
757 B
Markdown
# Activity 2
|
|
Define mathematically the mean square error of a position estimate. Explain and justify your answer.
|
|
|
|
The MSE is defined almost the same as variance, given by
|
|
$$
|
|
\frac{\sum{(x-\overline{x})^2}}{n}
|
|
$$
|
|
# Activity 3
|
|
Prove that the following holds for Gamma distribution:
|
|
$$
|
|
\alpha = \frac{\overline{x}^2}{s^2}
|
|
$$
|
|
$$
|
|
\beta = \frac{s^2}{\overline{x}}
|
|
$$
|
|
|
|
Which results by substition from the following:
|
|
$$ \overline{x} = \alpha \beta, \quad s^2 = \alpha \beta^2 $$
|
|
$$
|
|
\alpha \left(\frac{\overline{x}}{\alpha}\right)^2 = s ^2
|
|
$$
|
|
yielding the first definition for $\alpha$.
|
|
Substituting into the expression for $\beta$ results in
|
|
$$
|
|
\beta = \frac{\overline{x}}{\alpha} = \frac{\overline{x}}{\frac{\overline{x}^2}{s^2}} = \frac{s^2}{\overline{x}}
|
|
$$
|