expected value of a die: (1/6)*1 + (1/6)*2 + (1/6)*3 + ... (1/6)*6 = 7/2
expected value with large outcomes
The expected number of successes when n mutually independent Bernoulli trials are performed, where p is the probability of success on each trial, is np.
If random variable X has the geometric distribution with parameter p, then E(x) = 1/p. (expected value)
Expected value IRV
If X and Y are independent random variables on a sample space S, then E(XY) = E(X)E(Y).
If X is a random variable on a sample space S and E(X) = mu, then V(X) = E((X - mu)2)
V(X) = E(X2) - E(X)2.
Variance => how widely the values of expected value of a random value is distributed.
Variance of the number of successes in n Bernoulli trials is npq, where q is 1 - p
The likelihood that a random variable takes a value far from it's expected value. This inequality provides an upper bound on the probability that the value of a random variable differs from the expected value of the random variable by more than a specified amount.
If relations a and b are related by an equivalence relation, they are equivalent, denoted by a ~ b
Bipartite Rules for Special Simple Graphs
Strong vs Weak conn.
A directed graph is strongly connected if there is a path from a to b and from b to a whenever a and b are vertices in the graph.
A directed graph is weakly connected if there is a path between every two vertices in the underlying undirected graph.
Strongly connected graph is also weakly connected. Look for strong components (vertices and cycles)
Euler Circuit Rules for Spec. Graphs
A connected multigraph with at least two vertices has an Euler circuit if and only if each of its vertices has even degree. It has an Euler path but not a circuit if and only if it has exactly two vertices of odd degree.