1 Distributions of Random Variables
To establish uniform notation and to provide the reader with a convenient reference/vocabulary we will cite some of the concepts and formulae of probability theory and statistics that we will use later in this chapter.
1.1 Measures of the ‘location’ of a distribution
Expected value . Among the location parameters , designed to show where the ‘bulk’ of a distribution is concentrated, the expected value (or, as we will alternatively call it, the mean ) is considered as standard (provided that it exists). We will usually denote it by μ. If we want to make it clear that we are talking about the expected value of the random variable X, then we will use the notation E(X). Further notations used in the same sense are 〈X〉 and μ X .
(#1) Frequently used synonyms for the expected value/mean are expectation value , mathematical expectation or just expectationIn physics and other fields of science (except mathematics), the expected value is often referred to,...
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- 1.
1 The expression’ spectrum point’ refers to any of the ‘allowed’ values of a discrete random variable in this context, and therefore it has nothing to do with the nuclear spectra discussed in Section 5.
- 2.
2 The explanation of the adjective ‘differential’ is made clear by Eq. (6).
- 3.
3 If the density function or the mass function is referred to as ‘distribution function’, then the ‘real’ distribution function is normally called integral distribution function . The reason for the name is clear from Eq. (5).
- 4.
4 As a matter of fact, very often, no reference is made to any distribution at all, although careful analysis of the problem reveals that some of the quantities are un-normalized density functions. As an example we mention the various quantities - actually different types of joint density functions of multivariate distributions - all referred to by the same term neutron flux and denoted by the same symbol (either ϕ, φ or Φ no matter how many and which of the possible variables (space coordinates, solid angle, speed or energy or, alternatively, lethargy) are considered or made disappear by integration. (See, for instance, Chapter 1 in Volume 4 and Chapter 4 in Volume 5.) To make things even more confusing, the only related quantity ‘defined’ by IUPAP bears the name neutron flux density, (dimension: number of neutrons per square centimeter per second), although this is the least ‘density-function-like’ of the whole family of related quantities. Apropos dimension: when in doubt as regards what type of neutron flux we are encountered with, we propose dimensional analysis as a guide.
- 5.
5 As a matter of fact, the author himself has to concentrate very hard not to fall out of his chosen role as a ‘mathematician’, but he probably fails occasionally anyway.
- 6.
6 Note that physicists often use the term lifetime not only in the sense we do in this chapter, but also in the sense mean life. Fortunately, in the really important cases, i.e., when quantitative statements are made (e.g., ‘the lifetime of the radionuclide is 10 s'), the ambiguity is removed and the reader can be sure that such a statement actually refers to the mean life.
- 7.
7 Note that the expression’ spectrum point’ is used here in the same sense as in Section 5. See in contrast footnote 1 after Eq. (1).
- 8.
8 See remark (#14) for the recipe.
- 9.
9 The order r and the parameter ν are positive real numbers, and Γ(r) is the complete gamma function.
- 10.
10f01 and F01 are the N(0, 1) standard normal density function and distribution function, respectively.
- 11.
11 In physics and engineering etc. normal distribution is often called Gaussiandistribution. However, we will only use the expression Gaussian curve meaning an un-normalized normal distribution function as shown in FIGURE 16.
- 12.
12 The reason for the quotation marks is that with continuous distributions any x value has 0 probability. What this statement really means is that the probability dP = f(x)dx is at its maximum (dx is constant).
- 13.
13 In particle physics, the same function is called the Breit-Wigner curve (Lyons 1986). See also Chapter 2.
- 14.
14 The confrontation with this defectiveness often fails to occur. For instance, Bevington, in his often-cited work (Bevington 1969), writes about Cauchy distribution as if it had an expected value.
- 15.
15 Note that the density function is asymptotically proportional to x−2, and therefore the integrand in the expected value formula is proportional to x−1, hence the integral itself (∼ ln x) is boundless.
- 16.
16 For the generation of exponentially distributed random numbers see remark (#30).
- 17.
17 See the example mentioned in remark (#35).
- 18.
18 See remark (#24).
References to Statistical Aspects of Nuclear Measurements
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COHEN, E.R., GIACOMO, P., 1987, Physica A, 146, pp. 1–68.
FELLER, W., 1968, An Introduction to Probability Theory and its Applications, Vol. I, 3rd edition, (New York: John Wiley and Sons, Inc.).
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LEO, W.R., 1987, Techniques for Nuclear and Particle Physics Experiments (Berlin: Springer-Verlag), Chapter 4.
LUX, J., KOBLINGER, L., 1991, Monte Carlo Particle Transportation Methods: Neutron and Photon Calculations (Boston: CRC Press).
LYONS, L., 1986, Statistics for Nuclear and Particle Physicists (Cambridge: University Press).
OREAR, J., 1987, Notes on Statistics for Physicists. In Procedures Manual for the Evaluated Nuclear Structure Data File, edited by M.R. Bhat, (National Nuclear Data Center Brookhaven National Laboratory).
PRESS, W.H., TEUKOLSKY, S.A., VETTERLING, W.T., FLANNERY, B.P., 1999, Numerical Recipes in C — The Art of Scientific Computing, 2nd edition (Cambridge: University Press), Chapters 14–15.
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(2003). Statistical Aspects of Nuclear Measurements. In: Handbook of Nuclear Chemistry. Springer, Boston, MA. https://doi.org/10.1007/0-387-30682-X_7
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