ДискретнаяМатематика / Student Solutions Manual / chapter 8

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5.Let p1 denote the probability that any particular code symbol is erroneously transmitted through a communication system. Assume that on di erent symbols, errors occur independently of one another. Suppose also

that with probability p2, an erroneous symbol is corrected upon receipt. Let X denote the number of correct symbols in a message block consisting of n symbols (count after the correction process has been carried out). What is the probability distribution of X?

5. A symbol is transmitted correctly if either there is no problem in the transmission or it is transmitted incorrectly, but corrected upon receipt. This probability is 1 p1 + p1p2. We wish to nd the probability that i symbols out of n are transmitted correctly, or are corrected upon receipt. The probability distribution of X is a Bernoulli distribution with a success de ned as a symbol transmitted correctly (after corrections). Thus X(i) =

C(n; i)(1 p1 + p1p2)i(p1(1 p2))n i.

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9.Recall the 3 Satis ability Problem. We are given a set of m clauses, each of which contains three boolean variables connected together by OR's; the clauses are then connected together by AND's. For example, we might want to assign truth values to the variables X1; X2; : : : ; X5 so that the expression

(X1 OR X3 OR :X5) AN D (X1 OR :X2 OR X4)

is true. Here, there are only m = 2 clauses and n = 5 variables. In general, for n variables (not counting their complements separately), we might have to try as many as 2n truth assignments to the variables to determine whether the clauses can all be simultaneously satis ed. We can assume that no variable appears more than once in the same clause (whether complemented or not). If X appears with :X in the same clause, the clause is automatically satis ed. If X appears more than once in the same clause, that is redundant, so the clause does not truly contain three variables. (As an example of how probability theory can help solve a problem that would otherwise seem to require examining an exponentially large number of cases, consider the following idea. For each variable, toss a fair coin; if the coin comes up heads, set the variable to TRUE and its complement to FALSE; if it comes up tails, make the opposite assignment.) What is the expectation of the number of clauses that will be satis ed?

9. Let X be the random variable that counts the number of satis ed clauses in a formula . A clause is satis ed if at least one variable (including complements of variables) in it is TRUE. We wish to nd E(X). Consider a single clause (x1 _ x2 _ x3). There are 8 possible truth value assignments, 7 of which satisfy the clause. Thus a random assignment will satisfy the clause with probability 7=8. De ne m random variables X1; X2; ; Xm. Let Xi(!) = 1 if ! satis es the ith clause of , otherwise Xi(!) = 0. Here ! can be viewed as a bit string of length n that corresponds to a truth value assignment. If ! is chosen at random then P (Xi(!) = 1) = 7=8. We now calculate E(Xi). E(Xi(!)) = 0 P (Xi(!) = 0) + 1 P (Xi(!) = 1) = 7=8. Let X = X1 + X2 + + Xm. Now,

Pm

E(X) = i=1 E(Xi) = (7m)=(8).