Random walks and branching processes satisfy the Markov property, i.e. they are special cases or examples of Markov processes. Generating functions are one of the main tools for studying such processes. Similarly, the characteristic function (the Fourier transform of the probability density function) is one of the main tools for analyzing continuous distributions and random processes derived from these distributions.
a) Let X~N(1) and Y~N(2) be two independent Gaussian random variables. Let Z=X+Y. Using the characteristic functions of X, Y, and Z, prove that Z~N(3).
b) Let S=B*X and T=B*X^2, where B=1 uniformly and X, Y are defined in part a. Using the characteristic functions s=E[exp(i*s*S)] and t=E[exp(i*t*T)], prove that S and T are independent.
Let U=(U1, U2, ..., Us) be a random vector in R^s, where U is uniform in {-1, 0, 1}. The U's are assumed to be independent. Similarly, consider a vector V=(V1, V2, ..., Vs) with probability distribution identical to U. Also, U and V are independent. Denote by G(s) the probability generating function of U and V. Here we have G(s)=s^(-1)+1+s/3.
c) Find G(s), the probability generating function of U * V.
d) From G(s), get the probability generating function of the scalar product between U and V (U * V = U1*V1 + U2*V2 + ... + Us*Vs), and finally determine P(U * V = 0). This is the probability that the two random vectors U and V are orthogonal in dimension s. Note that G(s) is given by the expression: (256s^16 + 5120s^15 + 46848s^14 + 259840s^13 + 975968s^12 + 2627520s^11 + 5236336s^10 + 7869200s^9 + 9004545s^8 + 7869200s^7 + 5236336s^6 + 2627520s^5 + 975968s^4 + 259840s^3 + 46848s^2 + 5120s + 256) / (43046721s).
