# Differences of distributions inside Kalman filter.

I am studying the Kalman filter algorithm but i can't understand one point. The k factor has to be chosen in order to minimize the variance of the signal. This lead to following equation:

$k=\frac{\sigma^2_f}{\sigma^2_f+\sigma^2_o}$

where $\sigma^2_f$ is the variance of the forecast signal and $\sigma^2_o$ is the variance of the observed signal. I don't understand why they in general have to be different? Why $\sigma^2_f\neq \sigma^2_o$ ? Why the distributions of the two variables (forecast and observed) are different?

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Just a note on wording: "The $k$ factor" is more commonly known as the "Filter Gain". –  nbubis Apr 15 '12 at 11:03

I assume you mean a one-dimensional Kalman Filter, since otherwise the relation you stated is more complex (A covariance matrix instead of the scalar variance $\sigma$).