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EECS 203 Final Exam Cheat Sheet Cheat Sheet by

Final exam cheat sheet.

expected variable

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.

Geometric Distri­bution

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).
Theorem

Variance

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
 

Chebys­hev's Inequality

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.

Equiva­lence Relation

If relations a and b are related by an equiva­lence relation, they are equiva­lent, denoted by a ~ b

Graph Termin­ology

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Bipartite Rules for Special Simple Graphs

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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

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Euler circuit

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.
                                                                       
 

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