As mentioned above, Kalman filtering relies on two successive updates, namely the prediction step and the data update (also referred to as correction step). These steps are detailed below. During these updates, the goal is to update the mean m and covariance matrix of the state vector p (position and velocity here). First, let us introduce some variables mt-1: Expected value of p at time t, before observing z(t) (only 2(t-1)= [(1),..., 2(t-1)] have been observed). Ett-1: Covariance matrix of p at time t, before observing z(t) (only 2(t-1) = [z(1),..., 2(t-1)] have been observed). mtt: Expected value of p at time t after having observed z(t). Ett: Covariance matrix of p at time t, after having observed z(t). In other words, mt-1 and Ett-1 are respectively the mean and covariance matrix of f(pt) 2(t-1)). met and Ett are respectively the mean and covariance matrix associated with f(pt) |z(t)). To compute these quantities sequentially, we will use the Bayes' rule and compute marginal distributions. Let's assume that before any measurement, we model the prior distribution of p(0), denoted by f(p)), by a bivariate normal distribution with known mean mojo and known covariance matrix 20|0- First we will predict the value of p() (before observing z()) using the result of Question 3. Question 5: using f(p)) = f f(p) p)f(p0)dp(0), verify that f(p(1)) is the density of a bivariate Gaussian distribution and compute its mean and covariance matrix. Detail how you obtain the final result. (4 Marks) The prediction of p(t) (before observing z(t) but after having observed z(t-1)) can be per- formed in the similar fashion, as we will see next. Question 6: If we assume the f(p(t-1) 2(t-1)) is a Gaussian probability density function and using f(pt) 2(t-1)) = f(pt) |p(t-1))f (p(t-1) 2(t-1))dp(t-1), verify that f(pt) |z(t-1)) is