Probability and Statistics By Robert A. Crovelli1 Open ...

U.S. DEPARTMENT OF THE INTERIOR

U.S. GEOLOGICAL SURVEY

Probability and Statistics

for

Petroleum Resource Assessment

By

Robert A. Crovelli1

Open-File Report 93-582

This report is preliminary and has not been reviewed for conformity with U.S.

Geological Survey editorial standards. Any use of trade, product or firm names is for

descriptive purposes only and does not imply endorsement by the U.S. Government.

!U.S. Geological Survey, Box 25046, MS 971, DFC, Denver, Colorado 80225

1993

TABLE OF CONTENTS

I.

Venn diagram for describing the fields of probability and statistics ................

The relationship between probability and inferential statistics.........................

Probability..................................................................................................................

A. Basic Concepts....................................................................................................

1. Petroleum accumulation classification hierarchy...................................

2. Experiment, sample space, and event.......................................................

3. Venn diagram...............................................................................................

4. Tree diagram.................................................................................................

5. Event relations..............................................................................................

6. Combinatorial analysis (counting techniques)........................................

7. Definitions of probability............................................................................

8. Probability of event relations.....................................................................

9. Conditional probability...............................................................................

10. Probability rules...........................................................................................

11. Applications of probability rules...............................................................

12. Bayes1 rule.....................................................................................................

Page

1

2

3

4

4

5

6

6

7

8

9

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14

18

B. Random Variables and Probability Distributions.........................................

1. Discrete random variables..........................................................................

2. Discrete probability distributions..............................................................

3. Graphs of discrete probability distributions............................................

4. Continuous random variables ...................................................................

5. Continuous probability distributions.......................................................

6. Graphs of continuous probability distributions......................................

7. General graphs of continuous probability distributions........................

8. Monte Carlo simulation..............................................................................

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19

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26

C. Descriptive Parameters.....................................................................................

1. Measures of central location.......................................................................

a. Mean........................................................................................................

b. Median.....................................................................................................

c. Mode........................................................................................................

2. Mean, median, and mode related to skewness .......................................

3. Measures of variation..................................................................................

a. Variance...................................................................................................

b. Standard deviation................................................................................

4. Fractiles..........................................................................................................

5. Examples.......................................................................................................

6. LOGRAF........................................................................................................

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29

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32

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34

35

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39

D. Some Continuous Probability Distributions..................................................

1. Normal distribution.....................................................................................

2. PROBDIST model selection menu.............................................................

3. 7-fractile probability histogram.................................................................

4. 3-fractile probability histogram.................................................................

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44

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47

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

H.

Normal distribution (minimum/maxiinum)...........................................

Normal distribution (mean/standard deviation)...................................

Truncated normal distribution..................................................................

Lognormal distribution...............................................................................

Truncated lognormal distribution.............................................................

Exponential distribution.............................................................................

Truncated exponential distribution ..........................................................

Pareto distribution.......................................................................................

Truncated Pareto distribution....................................................................

Uniform distribution...................................................................................

Triangular distribution ...............................................................................

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52

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54

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58

Statistics ..................................................................................................................

A. Sampling Concepts............................................................................................

1. Populations...................................................................................................

2. Parameters....................................................................................................

3. Samples..........................................................................................................

4. Sampling techniques ...................................................................................

5. Statistics.........................................................................................................

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60

60

62

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71

B. Descriptive Statistics..........................................................................................

1. Tabular methods..........................................................................................

a. Frequency distribution..........................................................................

b. Relative frequency distribution...........................................................

c. Cumulative frequency distribution (less than)..................................

d. Relative cumulative frequency distribution (less than)...................

e. Cumulative frequency distribution (more than)...............................

f. Relative cumulative frequency distribution (more than).................

2. Pictorial methods.........................................................................................

a. Frequency histogram.............................................................................

b. Relative frequency histogram..............................................................

c. Cumulative frequency polygon (less than)........................................

d. Relative cumulative frequency polygon (less than).........................

e. Cumulative frequency polygon (more than).....................................

f. Relative cumulative frequency polygon (more than).......................

3. Measures of central location.......................................................................

a. Sample mean ..........................................................................................

b. Sample median.......................................................................................

c. Sample mode..........................................................................................

4. Measures of variation..................................................................................

a. Sample variance.....................................................................................

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b. Sample standard deviation...................................................................

89

c. Sample range..........................................................................................

90

C. Sampling Distributions.....................................................................................

1. Sampling distribution of the mean............................................................

2. Central Limit Theorem................................................................................

3. Normal probability paper...........................................................................

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ii

D. Inferential Statistics ...........................................................................................

1. Statistical estimation....................................................................................

a. Point estimation.....................................................................................

b. Interval estimation.................................................................................

2. Tests of hypotheses......................................................................................

a. Z test for p................................................................................................

b. Chi-squared goodness-of-fit test..........................................................

c. Lognormal probability paper...............................................................

3. Regression and correlation.........................................................................

a. formulas..................................................................................................

b. Transformations.....................................................................................

c. Finding-rate curves................................................................................

d. Power laws..............................................................................................

e. Fractals.....................................................................................................

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Ill

Ill

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Selected References............................................................................................................

Appendix: Tables..............................................................................................................

A.I. Areas under the normal curve................................................................................

A.2. Critical values of the chi-squared distribution.....................................................

Index.....................................................................................................................................

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136

111

Operations Research

Probability Models

Stochastic Processes

Markov Chains

Queuing Theory

Simulation

Game Theory

Probability

Decision Theory

Theory

Risk Analysis

Dynamic Programming

Reliability Theory

Combinatorial Analysis

Time Series Analysis

Actuarial Analysis

Random Walks

AREAS

Probability

Basic

Probability

Statistical

Theory

Statistical Inference

Estimation Theory

Tests of Hypotheses

Regression & Correlation

(Simple & Multiple)

Analysis of Variance

Design of Experiments

Sampling Techniques

Sample Surveys

Nonparametric Statistics

Multivariate Statistics

Factor Analysis

Discriminate Analysis

Quality Control

Descriptive Statistics

Bayesian Statistics

Geostatistics

AREAS

Statistics

Venn Diagram for Describing the Fields of Probability and Statistics

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