Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAP. 7] PROBABILITY 145 (ii) Two points, (1, 2), (2, 1), have sum 3; hencef( 3 )=^29. (iii) Three points, (1, 3), (2, 2), (1, 3 ...

146 PROBABILITY [CHAP. 7 (a) μ=xipi= 2 ( 1 3 ) + 3 ( 1 2 ) + 11 ( 1 6 ) = 4 E(X^2 )=x^2 ipi= 22 ( 1 3 ) + 32 ( 1 2 ) + 112 ( 1 ...

CHAP. 7] PROBABILITY 147 Thus the distributionhofZfollows: z 371115 h(z) 1 / 62 / 62 / 61 / 6 Hence E(Z)= ∑ zh(z)= 3 6 + 14 6 + ...

148 PROBABILITY [CHAP. 7 7.39. LetXbe a random variable with meanμ=40 and standard deviationσ=2. Use Chebyshev’s Inequality to f ...

CHAP. 7] PROBABILITY 149 SupplementaryProblems SAMPLE SPACES AND EVENTS 7.42. LetA,B, andCbe events. Rewrite each of the followi ...

150 PROBABILITY [CHAP. 7 7.54. Consider the following probability distribution: Outcomex 12345 ProbabilityP(x) 0.2 0.4 0.1 0.1 0 ...

CHAP. 7] PROBABILITY 151 REPEATED TRIALS, BINOMIAL DISTRIBUTION 7.67. Whenever horsesa,b, andcrace together, their respective pr ...

152 PROBABILITY [CHAP. 7 BINOMIAL DISTRIBUTION 7.83.The probability that a women hits a target isp= 1 /3. She fires 50 times. Fi ...

CHAP. 7] PROBABILITY 153 7.81. (a) [2, 6, 10, 12, 24, 36; 1/ 6 ,..., 1 /6]; (b)E(W)=15. No. 7.82. (a) 0.9, 1.09, 1.04; (b) (i) [ ...

CHAPTER 8 Graph Theory 8.1Introduction, Data Structures Graphs, directed graphs, trees and binary trees appear in many areas of ...

CHAP. 8] GRAPH THEORY 155 of pointers. Figure 8-1 is a schematic diagram of a linked list with six nodes. That is, each node is ...

156 GRAPH THEORY [CHAP. 8 Fig. 8-3 (b)Queue:Aqueue, also called afirst-in first-out(FIFO) system, is a linear list in which dele ...

CHAP. 8] GRAPH THEORY 157 Verticesuandvare said to beadjacentorneighborsif there is an edgee={u, v}. In such a case,uandv are ca ...

158 GRAPH THEORY [CHAP. 8 Finite Graphs, Trivial Graph A multigraph is said to befiniteif it has a finite number of vertices and ...

CHAP. 8] GRAPH THEORY 159 Fig. 8-7 8.4Paths, Connectivity A pathin a multigraphGconsists of an alternating sequence of vertices ...

160 GRAPH THEORY [CHAP. 8 Connectivity, Connected Components AgraphGisconnectedif there is a path between any two of its vertice ...

CHAP. 8] GRAPH THEORY 161 Fig. 8-10 We now show how Euler proved that the multigraph in Fig. 8-10(b)is not traversable and hence ...

162 GRAPH THEORY [CHAP. 8 Although it is clear that only connected graphs can be Hamiltonian, there is no simple criterion to te ...

CHAP. 8] GRAPH THEORY 163 Fig. 8-13 Fig. 8-14 Fig. 8-15 Bipartite Graphs A graphGis said to bebipartiteif its verticesVcan be pa ...

164 GRAPH THEORY [CHAP. 8 Fig. 8-16 8.8Tree Graphs A graphTis called atreeifTis connected andThas no cycles. Examples of trees a ...