For the following graphs, a. determine for which values of x = a the lim x → a f ( x ) exists but f is not continuous at x = a , and b. determine for which values of x = a the function is continuous but not differentiable at x = a . 79.
For the following graphs, a. determine for which values of x = a the lim x → a f ( x ) exists but f is not continuous at x = a , and b. determine for which values of x = a the function is continuous but not differentiable at x = a . 79.
28. (a) Under what conditions do we say that two random variables X and Y are
independent?
(b) Demonstrate that if X and Y are independent, then it follows that E(XY) =
E(X)E(Y);
(e) Show by a counter example that the converse of (ii) is not necessarily true.
7. [10 marks]
Let G = (V,E) be a 3-connected graph with at least 6 vertices. Let C be a cycle in G
of length 5. We show how to find a longer cycle in G.
(a) Let x be a vertex of G that is not on C. Show that there are three C-paths
Po, P1, P2 that are disjoint except at the shared initial vertex and only intersect
C at their final vertices.
(b) Show that at least two of P0, P1, P2 have final vertices that are adjacent along C.
(c) Combine two of Po, P1, P2 with C to produce a cycle in G that is longer than C.
1. Let X and Y be random variables and suppose that A = F. Prove that
Z XI(A)+YI(A) is a random variable.
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