1. (5점) 못쓸 경우 2점씩 감점    (5점) 다항식 차수늘려서 다 가능

2. (10점)

3. a) - f) 각 4점씩

4. 5/5/10점

5. 6/6/7점   a)3  b) 14 c) 7

6. 5점씩

Frequentist

The probability of an event E is the limit, as n→∞, of the fraction of times that V□E in n repetitions of the same experiment.

Problems:

Only well-defined for experiments that are infinitely repeatable (at least in principle).

Can never be measured exactly in finite time!

Advantage: Objective, mathematical def’n.

Bayesian

Suppose a rational entity R is offered a choice between two rewards:

Winning $1 if event E occurs.

Receiving p dollars (where p□[0,1]) unconditionally.

If R is indifferent between these two rewards, then we say R’s probability for E is p.

Problem: Subjective definition, depends on the reasoner R, and his knowledge & rationality.

Laplacian

First, assume that all outcomes in the sample space are equally likely

This term still needs to be defined.

Then, the probability of event E, Pr[E] = |E|/|S|.   Very simple!

Problems: Still needs a definition for equally likely, and depends on existence of a finite sample space with all equally likely outcomes.

Axiomatic

Let p be any function p:S→[0,1], such that:

0 ≤ p(s) ≤ 1 for all outcomes s□S.

∑ p(s) = 1.

Such a p is called a probability distribution.

Then, the probability of any event E□S is just:

Advantage: Totally mathematically well-defined.

Problem: Leaves operational def’n unspecified.

7. 각 4점

a)A directed graph (V,E) consists of a set of vertices V and a binary relation E on V.

b)The in-degree of v, deg□(v), is the number of edges going to v.

The out-degree of v, deg□(v), is the number of edges coming from v.

c)

d) The degree of v, deg(v), is its number of incident edges.

(Except that any self-loops are counted twice.)

e)

8. 각 5점

A binary relation R from A to B, written (with signature) R:A↔B, is a subset of A×B.

E.g., let < : N↔N :≡ {(n,m) | n < m}

a) In predicate logic, a predicate is modeled as a function P(·) from objects to

propositions. P(x) = “x is sleeping” (where x is any object).

xRb where b 0s {T,F}

b) If there is a function f from a set A to a set B, we can form a relation R by assigning a of A the unique element b in B that aRb. (Relations are a generalization of functions)

c) a symmetric binary relation corresponds to a simple graph.

d) directed multigraph (or directed pseudograph )