A First Course Mathematical Statistics

Front Cover
CUP Archive, Jan 2, 1949 - Mathematics - 271 pages
This book provides the mathematical foundations of statistics. Its aim is to explain the principles, to prove the formulae to give validity to the methods employed in the interpretation of statistical data. Many examples are included but, since the primary emphasis is on the underlying theory, it is of interest to students of a wide variety of subjects: biology, psychology, agriculture, economics, physics, chemistry, and (of course) mathematics.
 

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Contents

Chapter I
1
Change of origin and unit
4
Variance Standard deviation
6
Moments
8
Grouped distribution
10
Continuous distributions
12
EXAMPLES I
16
PROBABILITY AND PROBABILITY DISTRIBUTIONS 7 Explanation of terms Measure of probability
19
Gamma distribution and Gamma variates
149
Sum of independent Gamma variates
151
Beta distribution of the first kind
153
Alternative proof of theorems
154
Produot of a Bill m variate and a yl+m variate
156
Quotient of independent Gamma variates
158
EXAMPLES VIII
160
Chapter IX
164

Theorems of total and compound probability
21
Probability distributions Expected valuo
24
Expected value of a sum or a product of two variates
26
Repeated trials Binomial distribution
28
Continuous probability distributions
30
Theorems of Tchebychef and Bornoulli
32
Empirical definition of probability
34
Moment generating function and characteristic function
36
Cumulative function of a distribution
39
EXAMPLES II
42
Chapter III
46
Poissons distribution
47
Derivation from the binomial distribution page
50
Some properties of the normal distribution
51
Probabilities and relative frequencies for various intervals
55
Distribution of a sum of independent normal variates
57
EXAMPLES III
58
MATHEMATICAL NOTES
63
Chapter IV
67
Continuous distributions
68
Lines of regression
69
Coefficient of correlation Standard error of estimate
72
Estimates from tho regression equation
74
Change of units
75
Numerical illustration
76
Correlation of ranks
79
Bivariate probability distributions
80
Variance of a sum of variatos
82
EXAMPLES IV
83
FURTHER CORRELATION THEORY CURVED REGRESSION LINES 33 Arrays Linear regression
87
Correlation ratios
89
Calculation of correlation ratios
91
Other relations
92
Continuous distributions
93
Bivariate normal distribution
95
Intraclass correlation
97
Polynomial regression Normal equations page
99
Large samples Test of significance
111
Sampling distributions Standard orrors
117
Standard error of a partition value
124
Standard errors of class frequencies
131
Comparison of the standard deviations of two large samples
137
Sampling from a Bivariate Population 61 Sampling covariance of the means of the variables page
138
Variance and covariance of moments about a fixed point
140
Standard error of the covariance of a large sample
141
EXAMPLES VII
144
Chapter VIII
146
Relation between the two functions
147
Linear constraints Degrees of freedom
166
population
169
Nature of the chisquare test An illustration
170
Test of goodness of fit
173
Numerical examples
175
Additive property of chisquare
177
Distribution of regression coefficients and correlation ratios
179
EXAMPLES IX
181
FURTHER TESTS OF SIGNIFICANCE SMALL SAMPLES 84 Small samples page
185
Students Distribution 85 The statistic t and its distribution
186
Test for an assumed population mean
189
Comparison of the means of two samples
190
Significance of an observed correlation
192
Significance of an observed regression coefficient
194
Distribution of the range of a sample
195
Distribution of the Variance Ratio 92 Ratio of independent estimates of the population variance
196
Fishers z distribution Table of F
198
Fishers Transformation of the Correlation Coefficient 94 Distribution of r Fishers transformation
200
Comparison of correlations in independent samples
202
Combination of estimates of a correlation coefficient
203
EXAMPLES X
205
Chapter XI
209
Homogeneous population One criterion of classification
210
Calculation of tho sums of squares
212
Two oriteria of classification
214
The Latin square Three oritoria of classification
217
Significanco of an observed correlation ratio
221
Significance of a regression funotion
223
Test for nonlinearity of regression
224
Resolution of the sum of products One criterion of classification page
226
Calculation of the sums of products
228
Examination and elimination of the effect of regression
229
Two criteria of classification
233
EXAMPLES XI
236
Chapter XII
242
Introductory Yules notation 242 109 Introductory Yules notation 110 Distribution of three or more variables
244
Determination of the coefficients of regression
246
Multiple correlation
249
Partial correlation
250
Reduction formula for the order of a standard deviation
253
Reduction formula for the order of a regression coefficient
254
Normal distribution
255
Significance of an observed partial correlation
256
Significance of an observed multiple correlation
257
EXAMPLES XII
260
LITERATURE FOR REFERENA
263
INDEX
273

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