EEL6825: HW#1

Due Friday, September 11, 1998 in class. Do not be late to class. Do not hand in any matlab code or plots-instructions for turning in matlab code will be discussed in class.

1.

Three one-dimensional distributions are given as uniform in [-1/3,1/3] for $\omega_1$, uniform in [-1/2,1/2] for $\omega_2$ and uniform in [-1,1] for $\omega_3$. $P(\omega_1)=P(\omega_2)=P(\omega_3)$.

(a)

Compute $p(\omega_i\vert x)$ for each class and sketch each function on a separate plot.

(b)

Consider a Bayes classifier for the three distributions. Be sure to describe the class for each possible value of x.

2.

2.5 in DH&S

3.

2.6 in DH&S

4.

2.10 in DH&S

5.

Two normal distributions are characterized by:

$$P(\omega_1)=P(\omega_2)$$

$$\mu_1=\mu_2= \left[ \begin{array}{c} 0 \\ 0 \end{array}\right]$$

$$\Sigma_1= \left[ \begin{array}{cc} 1&0 \\ 0&1/4 \end{array}\right]$$

$$\Sigma_2= \left[ \begin{array}{cc} 1/4&0 \\ 0&1 \end{array}\right]$$

(a)

Plot the contours of constant d²(X) (constant Mahalanobis distance) for each distribution by setting d²(X)=1for each contour. Clearly label each contour.

(b)

Compute the Bayes classifier (g(x)).

(c)

Sketch the Bayes Classifier on a graph that also shows probability contours. Clearly label the classification regions.

6.

Computer Exercise 2.7 in DH&S