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#### probability set up

Cornell, BTRY 4080

Excerpt: ... n Ai . i=1 An exercise is to show that (n Ai )c = n Ac and (n Ai )c = n Ac . These are called DeMori=1 i=1 i i=1 i=1 i gan's laws. There are no restrictions on S. The collection of events, F, must be a -field, which means that it satisfies the following: (i) , S is in F; (ii) if A is in F, then Ac is in F; (iii) if A1 , A2 , . . . are in F, then Ai and Ai are in F. i=1 i=1 Typically we will take F to be all subsets of S, and so (i)-(iii) are automatically satisfied. The only times we won't have F be all subsets is for technical reasons or when we talk about conditional expectation. So now we have a space S, a -field F, and we need to talk about what a probability is. There are three axioms: (1) 0 P(E) 1 for all events E. (2) P(S) = 1. (3) If the Ei are pairwise disjoint , P( Ei ) = i=1 i=1 P(Ei ). Pairwise disjoint means that Ei Ej = unless i = j. Note that probabilities are probabilities of subsets of S, not of points of S. However it is common to write P(x) for P({x}). Intuitively, the probability ...

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#### hw5

UNL, SHARTKE 2

Excerpt: ... or two adjacent vertices. (Hint: Study s(u) - s(v) when u and v are adjacent.) b) Determine the maximum distance between the center and the barycenter in a tree of diameter d. (Example: In the tree below, the center is {x, y}, the barycenter is {z}, and the distance between them is 1.) x y z 3. Let G be the 3-regular graph with 4m vertices formed from m pairwise disjoint kites by adding m edges to link them in a ring, as shown on the right above for m = 6. Prove that (G) = 2m8m . 4. Count the following sets of trees with vertex set [n], giving two proofs for each: one using the Prfer correspondence and one by direct counting arguments. u a) trees that have 2 leaves. b) trees that have n - 2 leaves. 5. Prove that if the Graceful Tree Conjecture is true and T is a tree with m edges, then K2m decomposes into 2m - 1 copies of T . (Hint: If every tree T with m - 1 edges is graceful, then K2m-1 has a cyclically invariant decomposition into copie ...

• 2 Pages