CH5115: Parameter and State Estimation Assignment 1

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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)?