# Unscented Kalman Filter Equations for Constant Turn Rate and Velocity Process Model

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.

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?

In the 1st order discretization we use Zero Hold filters, so if there is noise add at time $${t}_{0}$$ it will effect the model within the time interval. So in the piece wise constant model it means its effect is the value multiplied by the the time interval of the model.