Mathematical Methods for Physics and Engineering - Matematica.NET

30.6 FUNCTIONS OF RANDOM VARIABLESYy + dyydx 1 dx 2XFigure 30.9 Illustration of a function Y (X) whose inverse X(Y ) is a doublevaluedfunction of Y . The range y to y + dy corresponds to X being either inthe range x 1 to x 1 + dx 1 or in the range x 2 to x 2 + dx 2 .This result may be generalised straightforwardly to the case where the range y toy + dy corresponds to more than two x-intervals.◮The random variable X is Gaussian distributed (see subsection 30.9.1) with mean µ andvariance σ 2 . Find the PDF of the new variable Y =(X − µ) 2 /σ 2 .It is clear that X(Y ) is a double-valued function of Y . However, in this case, it isstraightforward to obtain single-valued functions giving the two values of x that correspondto a given value of y; thesearex 1 = µ − σ √ y and x 2 = µ + σ √ y,where √ y is taken tomean the positive square root.The PDF of X is given byf(x) = 1 [ ]σ √ 2π exp (x − µ)2− .2σ 2Since dx 1 /dy = −σ/(2 √ y)anddx 2 /dy = σ/(2 √ y), from (30.59) we obtaing(y) = 1∣ ∣∣∣σ √ 2π exp(− 1 y) −σ22 √ y ∣ + 1∣ ∣∣∣σ √ 2π exp(− 1 y) σ22 √ y ∣= 12 √ π ( 1 2 y)−1/2 exp(− 1 y). 2As we shall see in subsection 30.9.3, this is the gamma distribution γ( 1 , 1 ). ◭ 2 230.6.3 Functions of several random variablesWe may extend our discussion further, to the case in which the new randomvariable is a function of several other random variables. For definiteness, let usconsider the random variable Z = Z(X,Y ), which is a function of two otherRVs X and Y . Given that these variables are described by the joint probabilitydensity function f(x, y), we wish to find the probability density function p(z) ofthe variable Z.1153