Book contents
- Frontmatter
- Contents
- Preface
- 1 Fourier's representation for functions on, Tp, and ℙN
- 2 Convolution of functions on, Tp, and ℙN
- 3 The calculus for finding Fourier transforms of functions on ℝ
- 4 The calculus for finding Fourier transforms of functions on Tp, and ℙN
- 5 Operator identities associated with Fourier analysis
- 6 The fast Fourier transform
- 7 Generalized functions on ℝ
- 8 Sampling
- 9 Partial differential equations
- 10 Wavelets
- 11 Musical tones
- 12 Probability
- Appendices
- Index
12 - Probability
Published online by Cambridge University Press: 01 September 2010
- Frontmatter
- Contents
- Preface
- 1 Fourier's representation for functions on, Tp, and ℙN
- 2 Convolution of functions on, Tp, and ℙN
- 3 The calculus for finding Fourier transforms of functions on ℝ
- 4 The calculus for finding Fourier transforms of functions on Tp, and ℙN
- 5 Operator identities associated with Fourier analysis
- 6 The fast Fourier transform
- 7 Generalized functions on ℝ
- 8 Sampling
- 9 Partial differential equations
- 10 Wavelets
- 11 Musical tones
- 12 Probability
- Appendices
- Index
Summary
Probability density functions on ℝ
Introduction
A liter bottle filled with air (at room temperature and pressure) holds approximately 2.7 · 1022 molecules of N2, O2, CO2, … that fly about at high-speed colliding with one another and with the walls of the bottle. We cannot hope to keep track of every particle in such a large ensemble, but in the mid-19th century, Maxwell showed that we can deduce many statistical properties about such a system. For example, he found the probability density
for the speed V of a given gas molecule, see Fig. 12.1. (The parameter
depends on the mass m of the molecule, the absolute temperature T and Boltzmann's constant k = 1.3806 … · 10-23 joule/K.) We use the integral
to find the probability that a molecule will have a speed in the interval v1 < V < v2.
Example Use Maxwell's density to find the fraction of N2 molecules (with m = 4.65 · 10-26 kg) in a warm room (with T = 300 K) that have speeds less than v0 = 1000 km/hr = 278 m/sec.
Solution We use (1) with the parameter
and the Maclaurin series for the exponential to compute
Approximately one-sixth of the N2 molecules have speeds less than 1000 km/hr! ▪
We can use the density function (1) to compute the average (or expected) value of certain functions of the molecular speed V.
Example Use Maxwell's density to find the average speed and the average kinetic energy for an N2 molecule when T = 300 K.
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- A First Course in Fourier Analysis , pp. 737 - 798Publisher: Cambridge University PressPrint publication year: 2008