Mathematical Methods for Physics and Engineering - Matematica.NET

30.6 FUNCTIONS OF RANDOM VARIABLESwhere the range of integration is over all possible values of the variables x i .Thisintegral is most readily evaluated by substituting in (30.62) the Fourier integralrepresentation of the Dirac delta function discussed in subsection 13.1.4, namelyδ(Z(x 1 ,x 2 ,...,x n ) − z) = 1 ∫ ∞e ik(Z(x1,x2,...,xn)−z) dk. (30.63)2π −∞This is best illustrated by considering a specific example.◮A general one-dimensional random walk consists of n independent steps, each of whichcan be of a different length and in either direction along the x-axis. If g(x) is the PDF forthe (positive or negative) displacement X along the x-axis achieved in a single step, obtainan expression for the PDF of the total displacement S after n steps.The total displacement S is simply the algebraic sum of the displacements X i achieved ineach of the n steps, so thatS = X 1 + X 2 + ···+ X n .Since the random variables X i are independent and have the same PDF g(x), their jointPDF is simply g(x 1 )g(x 2 ) ···g(x n ). Substituting this into (30.62), together with (30.63), weobtain∫ ∞ ∫ ∞ ∫ ∞p(s) = ··· g(x 1 )g(x 2 ) ···g(x n ) 1 ∫ ∞e ik[(x 1+x 2 +···+x n)−s] dk dx 1 dx 2 ···dx n2π= 12π−∞ −∞∫ ∞−∞−∞−∞(∫ ∞ndk e −iks g(x)e dx) ikx . (30.64)−∞It is convenient to define the characteristic function C(k) of the variable X asC(k) =∫ ∞−∞g(x)e ikx dx,which is simply related to the Fourier transform of g(x). Then (30.64) may be written asp(s) = 1 ∫ ∞e −iks [C(k)] n dk.2π −∞Thus p(s) can be found by evaluating two Fourier integrals. Characteristic functions willbe discussed in more detail in subsection 30.7.3. ◭30.6.4 Expectation values and variancesIn some cases, one is interested only in the expectation value or the varianceof the new variable Z rather than in its full probability density function. Fordefiniteness, let us consider the random variable Z = Z(X,Y ), which is a functionof two RVs X and Y with a known joint distribution f(x, y); the results we willobtain are readily generalised to more (or fewer) variables.It is clear that E[Z] andV [Z] can be obtained, in principle, by first using themethods discussed above to obtain p(z) and then evaluating the appropriate sumsor integrals. The intermediate step of calculating p(z) is not necessary, however,since it is straightforward to obtain expressions for E[Z] andV [Z] in terms of1155