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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 ...
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- Winter '10
- Probability, Communications system