### expected variableexpected value of a die: (1/6)*1 + (1/6)*2 + (1/6)*3 + ... (1/6)*6 = 7/2 ### expected value with large outcomesThe expected number of successes when n mutually independent Bernoulli trials are performed, where p is the probability of success on each trial, is np. ### Geometric DistributionIf random variable X has the geometric distribution with parameter p, then E(x) = 1/p. (expected value) ### Expected value IRVIf X and Y are independent random variables on a sample space S, then E(XY) = E(X)E(Y). |
### VarianceIf X is a random variable on a sample space S and E(X) = mu, then V(X) = E((X - mu)^{2})
V(X) = E(X^{2}) - 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* | | ### Chebyshev's InequalityThe 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.
### Equivalence RelationIf 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### Euler circuitA 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. |

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