# [ECON 4] - Midterm Exam Guide - Everything you need to know! (19 pages long)

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ECON 4

MIDTERM EXAM

STUDY GUIDE

Econ 41 (Fall 2015)

Department of Economics, UCLA

Instructor: Shuyang Sheng

Lecture 1: Probability: Some Basic Concepts

1 Probability

1.1 Basic Concepts

De…nition 1 In the study of statistics, we consider experiments for which the outcome can-

not be predicted with certainty. Such experiments are called random experiments.

De…nition 2 Usually experiments are such that the collection of every possible outcome can

be described and listed. The collection of all outcomes is called the outcome space, and is

denoted by O.

Example 3 Roll one die and count the number of the upper face. In this example, the

outcome space O=f1;2;3;4;5;6g.

Example 4 Roll two dies and count the sum of the numbers of the upper faces. In this

example, the outcome space O=f2;3;4;5;6;7;8;9;10;11;12g.

Example 5 Flip a coin at random and …nd the picture on the upper face. In this example,

the outcome space O=fhead, tailg.

Example 6 A fair coin is ‡ipped successively until the same face is observed on successive

‡ips. In this random experiment, we have

O=fHH; T T; T HH; HT T; HT HH; T HT T; T HT HH; HT HT T; :::::g

which implies there are in…nite many elements/possible outcomes in O.

De…nition 7 Given an outcome space O, let Abe a part of the collection of outcomes in O,

that is, AO. Then Ais called an event. When a random experiment is performed and

the outcome of the experiment is in A, we say that event A has occured.

Example 8 Roll one die. O=f1;2;3;4;5;6g.A=f1;2;3g. Or A=f2;4;6g.

1

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Review of algebra of sets It is useful because, in studying probability, the words sets

and events are interchangeable.

?denotes the null or empty set.

BAmeans that Bis a subset of A.

A[Bis the union of Aand B.

A\Bis the intersection of Aand B.

A0is the complement of A, i.e., all the elements of Othat are not in A.

Example 9 Roll one die. O=f1;2;3;4;5;6g.A=f2gand B=f1;2;3g. Then AB,

A[B=f1;2;3g=B,A\B=f2g=Aand A0=f1;3;4;5;6g.

Special terminologies used by statisticians

A1and A2are mutually exclusive events if A1\A2=?, i.e., if they are disjoints.

Likewise, we say that A1; A2; A3are mutually exclusive events if A1\A2=?,A2\A3=

?,A3\A1=?. In general, we say that A1; A2; : : : ; Akare mutually exclusive events

if Ai\Aj=?for i6=j.

A1; A2; : : : ; Akare exhaustive events if A1[A2[: : : [Ak=O.

If A1; A2; : : : ; Akare mutually exclusive and exhaustive events, then Ai\Aj=?for

i6=jand A1[A2[: : : [Ak=O.

The mutually exclusive and exhaustive events can be extended to the case with in…nite

but countable many sets, i.e. Ai(i= 1;2;3; :::) are mutually exclusive and exhaustive

if Ai\Aj=?for i6=jand A1[A2[::: =[1

i=1Ai=O.

Intuitive de…nition of probability

The probability of Ais usually written as P(A).

Consider repeating an experiment ntimes. Count the number of times that event A

actually occurred. This number is called the frequency of event Aand is sometimes

written N(A).

The ratio N(A)/ nis called the relative frequency of event A.

A relative frequency is usually very unstable for small values of n, but it tends to

stabilize as nincreases. If it does stabilize around p, say, then we say that P(A) = p.

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## Document Summary

De(cid:133)nition 1 in the study of statistics, we consider experiments for which the outcome can- not be predicted with certainty. De(cid:133)nition 2 usually experiments are such that the collection of every possible outcome can be described and listed. The collection of all outcomes is called the outcome space, and is denoted by o. Example 3 roll one die and count the number of the upper face. outcome space o = f1; 2; 3; 4; 5; 6g. Example 4 roll two dies and count the sum of the numbers of the upper faces. In this example, the outcome space o = f2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12g. Example 5 flip a coin at random and (cid:133)nd the picture on the upper face. In this example, the outcome space o = fhead, tailg. Example 6 a fair coin is (cid:135)ipped successively until the same face is observed on successive (cid:135)ips.