ECE 250 Handout # 4

UCSD ECE 250 Handout # 4

ECE 250 Handout #

Solutions to Exercise Set #4 1. Optical communication channel. Let the signal input to an optical channel be given by X = 1 with probability 1 2 10 with probability 1 2. Find y * and the corresponding probability of error.

ECE 250 Handout # 27

1. Drawing balls without replacement (20 pts). Suppose that we have an urn containing one red ball and n − 1 white balls. Each time we draw a ball at random from the urn without replacement (so after the n-th drawing, there is no ball left in the urn). X i = 1 if the i-th ball is red, 0 otherwise.

UCSD ECE 250 Handout # 18

(a) Find P{X10 = 10}. (b) Approximate P{−10 ≤ X100 ≤ 10} using the central limit theorem. (c) Find P{Xn = k}. 2. Absolute-value random walk. Consider the symmetric random walk Xn in the previous problem. Define the absolute value random process Yn = |Xn|. (a) Find P{Yn = k}. (b) Find P{max1≤i<20 Yi = 10 |Y20 = 0}. 3. A random process. Let Xn = Zn−1 + Zn for n ≥ 1, where Z0, Z1, Z2, . . . are i....

UCSD ECE 250 Handout # 16

4. Noise cancellation. A classical problem in statistical signal processing involves estimating a weak signal (e.g., the heart beat of a fetus) in the presence of a strong interference (the heart beat of its mother) by making two observations; one with the weak signal present and one without (by placing one microphone on the mother’s belly and another close to her heart). The observations can t...