assignment2

assignment2 - assignment 2 1 Show that n k = n n−k 2 Show...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: assignment 2 1. Show that n k = n n−k 2. Show that if P [A ∩ B ∩ C ] = P [A | B ∩ C ]P [B | C ]P [C ]. 3. A nonsymmetric binary communications channel is shown in the ﬁgure below. Assume the input is “0” with probability p and “1” with probability 1 − p. (a) Find the probability that the out put is 0. (b) Find the probability that the input was 0 given that the output is 1. Find the probability that the input is 1 given that the output is 1. Which input is more probable? Input 0 bb /0 Ñ@ bbε1 Ñ bb ÑÑ bb ÑÑÑ b Ñ ÑÑ bbb ε2 ÑÑ bb Ñ b ÑÑ Ñ 1−ε2 b /1 1 1−ε1 Output 4. A ternary communication system is shown in the ﬁgure below. Suppose that input symbols 0, 1 and 2 occur with probability 1/3 respectively. (a) Find the probabilities of the output symbols. (b) Suppose that a 1 is observed at the output. What is the probability that the input was 0? 1? 2? 1 I nput 0 vvvε /0 ÔA vvv vvv ÔÔÔ Ô Ôvvvv 1−ε ÔÔ &/ 1 1 wwwε ÔÔÔ wÔ w Ôw ww ε ÔÔ www ww& ÔÔ /2 2 1−ε 1−ε Output 5. In the binary communication system in attached example, ﬁnd the value of ε for which the input of the channel is independent of the output of the channel. Can such a channel be used to transmit information? Attached Example: Binary Communication System Many communication systems can be modeled in the following way. First, the user inputs a 0 or a 1 into the system, and a corresponding signal is transmitted. Second, the receiver makes a decision about what was the input to the system, based on the signal it received. Suppose that the user sends 0s with probability 1 − p and 1s with probability p, and suppose that the receiver makes random decision errors with probability ε. For i = 0, 1, let Ai be the event ”input was i,” and let Bi be the event ”receiver decision was i.” Find the probabilities P [Ai ∩ Bj ] for i = 0, 1 and j = 0, 1. Then we have P [A0 ∩ B0 ] = (1 − p)(1 − ε) P [A0 ∩ B1 ] = (1 − p)ε P [A1 ∩ B0 ] = pε P [A1 ∩ B1 ] = p(1 − ε) 2 ...
View Full Document

{[ snackBarMessage ]}

Page1 / 2

assignment2 - assignment 2 1 Show that n k = n n−k 2 Show...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online