Discrete Mathematics for Computer Science

Exercises 517 Hence, from Bayes' Theorem and the Theorem of Total Probability, P(A 2 I B) = P(B IAD) P(A) P(B) P(B I A 2 ). P(A ...

518 CHAPTER 8 Discrete Probability Suppose that E 1 , E 2 ... , Ek are events in the same sample space and that some pair Ei, E ...

Exercises 519 (a) Suppose you decide to stand with your original choice. What are your chances of winning the car? (b) Suppose y ...

520 CHAPTER 8 Discrete Probability U Discrete Random Variables Many probabilistic experiments have outcomes that are associated ...

Discrete Random Variables 521 The term random variable is unfortunate, because X is not a variable in the mathe- matical sense a ...

522 CHAPTER 8 Discrete Probability 8.72 The Binomial Distribution A distribution is simply a density that arises in a special wa ...

Discrete Random Variables 523 Hence, X is a function from 02 to {0, 1, 2. n}. The probability in Q? of the event (X = k) is comp ...

524 CHAPTER 8 Discrete Probability is given by b(0; 10, 25/1000) = C(10, 0) • (25/1000)0 .(1 - 25/1000)1° = (975/1000)'° S0.7763 ...

Discrete Random Variables 525 spots is, respectively, 1 1 1 1 1 1 6' 2' 12' 12' 12' 12 These probabilities define a random varia ...

526 CHAPTER 8 Discrete Probability Substituting px(x) = 2 p(w), where the summation is over all a) E? in the event (X = x), give ...

Discrete Random Variables 527 lands heads or tails, respectively. For a triple w E 2, let X 1 (w) be 1 if the first coin lands h ...

528 CHAPTER 8 Discrete Probability The scaling of a random variable. Given random variable X and a real number k, the random var ...

Exercises 529 (c) From part (b), with X =(XI + X 2 + + X.) and k = n, we have that E(X1 + X2-+Xn) n' nE(X + X 2 + ' + X.) so by ...

530 CHAPTER 8 Discrete Probability Repeat Exercise 3 for a coin that comes up heads a third of the time. 5. What is the relatio ...

Variance, Standard Deviation, and the Law of Averages 531 informally as the Law of Averages. The theorem says that under the rig ...

532 CHAPTER 8 Discrete Probability Regarding the random variable (X - /L)2 as the sum of the random variable X^2 , the random va ...

Variance, Standard Deviation, and the Law of Averages 533 Proof. By definition, or-2 = T-•(x(o) _- t)." P(CO) Since the summands ...

534 CHAPTER 8 Discrete Probability form an independent set. In other words (recall Definition 2 of Section 8.5.1: P(X 1 = X 1 , ...

Variance, Standard Deviation, and the Law of Averages 535 Properties (b) and (c) imply that i.i.d. random variables share the sa ...

536 CHAPTER 8 Discrete Probability Since there may be several pairs of values x1 e Qx 1 , and x2 E Qx 2 such that x1 • X2 = X, w ...