The Chemistry Maths Book, Second Edition

610 Chapter 21Probability and statistics systems at very low temperatures and, for example, for the description of the propertie ...

21.6 Permutations and combinations 611 Equation (21.26) is already very accurate whenn 1 = 152 , and can be assumed to be exact ...

612 Chapter 21Probability and statistics numbers (treated as continuous variables) subject to the constraints that the total num ...

21.7 Continuous distributions 613 21.7 Continuous distributions The distributions considered so far have been discrete, involvin ...

614 Chapter 21Probability and statistics with mean (expectation value) (21.37) and standard deviation given by (21.38) An exampl ...

21.8 The Gaussian distribution 615 If the variables are transformed to spherical polar coordinates in the space then is the prob ...

616 Chapter 21Probability and statistics The function is symmetric aboutx 1 = 1 μ. It is broad when σis large and narrow when σi ...

21.8 The Gaussian distribution 617 so thatF(a)is the probability that the variable has valuex 1 < 1 a, and (21.41) for unit t ...

618 Chapter 21Probability and statistics 21.9 More than one variable In many experiments in the physical sciences, the outcomes ...

21.10 Least squares 619 The first three are for the performance of a class of first-year chemistry students. Figure (a) shows th ...

620 Chapter 21Probability and statistics Simple least squares fitting The simplest and one of the most popular methods of fittin ...

21.10 Least squares 621 Division by then gives the pair of normal equations (21.53) F1− 1 mE1− 1 c 1 = 10 and these have solutio ...

622 Chapter 21Probability and statistics This is the example shown in Figure 21.10. The error bars correspond toσ 1 = 1 0.25so t ...

21.11 Sample statistics 623 is the probability density for the errorε i 1 = 1 y i 1 − 1 f(x i ). If the errors are independent, ...

624 Chapter 21Probability and statistics Considerations of ‘best estimates’ in sampling theory tell us that although (21.62) is ...

21.12 Exercises 625 6.For the data in Exercise 1,(i) calculate and , (ii)use equations (21.7) and (21.9) to compute the standard ...

626 Chapter 21Probability and statistics Section 21.7 23.The variable xcan have any value in the continuous range 01 ≤ 1 x 1 ≤ 1 ...

Appendix. Standard integrals Indefinite Integrals For simplicity, the constant of integration has been omitted from the tabulati ...

628 AppendixStandard integrals (a 1 ≠ 1 ±b) (a 1 ≠ 1 ±b) (a 1 ≠ 1 ±b) Ztanhxdx=lncoshx Zcoshxdx=sinhx Zsinhxdx=coshx Z dx x x 12 ...

Definite integrals 629 Z dx abxx xb +− ba = −         − 2 1 2 2 4 sin =++++           lnx b abxx 2 2 Z dx ...