SOLVED CH5115: Parameter and State Estimation Assignment 1

  1. (a) If two random variables have joint density f(x, y) =    K e −x/ye −y y x > 0, y > 0 0 elsewhere Find (i) the value of K (ii) marginal density of Y , (iii) the probability Pr(0 < X < 1, 0.2 < Y < 0.4) (iv) conditional expectation E(X|Y ). Use numerical integration routines (integral or integral2 in Matlab) if necessary. (b) Show that for two RVs X and Y that have a joint Gaussian distribution, the conditional expectation E(Y |X) is a linear function of X. 2. The covariance between two RVs is estimated from their samples x[k] and y[k] as σˆyx = 1 N X N k=1 (y[k] − y¯)(x[k] − x¯) (1) where x¯ and y¯ are the sample means of X and Y , respectively and N is the sample size. Write a function in Matlab to calculate this sample covariance matrix given samples of two random variables. Test your code on the case X ∼ N (1, 2) and Y = 3X2 + 5X by comparing the resulting covariance matrix with the values obtained from cov command in Matlab. Finally, show by means of simulation that the estimate σˆyx tends to the theoretical value as N → ∞. 3. Given the variance-covariance matrix of three random variables X1, X2 and X3, Σ =    4 1 2 1 9 −3 2 −3 25   , (a) Find the correlation matrix ρ. (b) Find the correlation between X1 and 1 2X2 + 1 2X3. 4. (a) Determine the optimal MAE predictor of a random variable X ∼ χ 2 (10), numerically using Matlab. Find the average absolute error at the optimum value X? . (b) Determine Pr(0.9X? < X < 1.1X? ). Is this lower than Pr(0.9µX < X < 1.1µX)? Justify your observation.