7.2 Sample values of Gaussian random variables Let Z be a Rayleigh random
varible (cf. Chapter 3, Example 3) with pdf
a. Show that a sample value, z, of A can be obtained by setting where r, as usual, denotes a random number (0 r I).
In Chapter 3, Example 3, it was shown that if S and Tare independent
Gaussian random variables with zero mean and standard deviation equal to
that is, with
pdf's
then the random variable Z = has the Rayleigh Of shown above (the parameter is equal to the standard deviations of S and T).
b. Using this fact and the result of part (a), show that if X and Y
are independent Gaussian random variables with mean m_{x} and m_{y},
respectively, and equal standard deviations,
then sample values, x and y, of X and Y can
be obtained simultaneously by setting
where r_{1} and r_{2} are independent random numbers in the interval [0, 1]. This result has already been quoted [cf. expressions (7.13)(7.15)]. Hint: Review carefully Example 3 of Chapter 3.
