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I am learning about Unscented Kalman Filters in Udacity's Self-Driving Car Nanodegree. The UKF is designed to track an object moving under the assumptions of constant turn rate $\ddot\psi$ and velocity(speed) $v$, the so-called CTRV model. The process noise is assumed to come from a longitudinal acceleration component $\nu_a = {N(0,\sigma_a^2)}$ and a yaw acceleration component $\nu_{\ddot\psi} = N(0,\sigma_{\ddot\psi}^2)$.

The mean state vector $$x = \begin{bmatrix} p_x \\ p_y \\ v \\ \psi \\ \dot\psi \\ \end{bmatrix}$$ tracks the mean values of x and y coordinates, speed, turn angle and the turn rate. The matrix $P$ stores the covariances of these variables.

The screenshots describing the model are below. Unscented Kalman Filter enter image description here

My understanding is that since the noise components are zero-mean, they should only affect the covariances not the means. However in the lecture noise is being added to the means too. For example $e=\Delta t \cdot \nu_{\ddot\psi}$ is the effect of of the yaw acceleration noise on the yaw rate. Also the notation seems odd. What does it mean to multiply by $\Delta t$ a stochastic variable $\nu_{\ddot\psi}$ with a mean and variance components?

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