(7) [40 marks| Generating random graphs is an important primitive in graph theory. A simple way to generate undirected random graphs with n nodes is to connect each pair of distinct nodes (self-loops are not allowed) independently with probability p. e For n nodes, if X is the r.v. equal to the number of edges in a randomly generated undirected graph, specify the PDF x. [10 marks| e What is the expected number of edges in a random graph? [5 marks| e The degree of a node is the number of other distinct nodes that it is connected to. Consider a node ue{1,2,..., n} and define Y, to be the r.v. equal to its degree in a random graph. When n is even, calculate the Pr [Yu_ > g] 7 |10 marks]| e A triangle in a graph is a set of 3 distinct nodes that are all connected to each other. For example, if there are edges from node u to v, v to w and w to wu, then nodes {u,v, w} are said to form a triangle. If Z is the r.v. equal to the number of triangles in a random graph, calculate E[Z]. [15 marks|