A town starts a mosquito control program and the rv Zn is the number of mosquitoes at the end of the nth year (n = 0, 1, 2, .. .). Let Xn be the growth rate of the mosquito population in year n; i.e., Zn = XnZn-1; n ≥ 1. Assume that {Xn;n ≥ 1} is a sequence of IID rv s with the PMF Pr{X=2} = 1/2; Pr{X=1/2} = 1/4; Pr{X=1/4} = 1/4. Suppose that Z0, the initial number of mosquitoes, is some known constant and assume for simplicity and consistency that Zn can take on non-integer values.

(a)   Find E [Zn] as a function of n and find limn→∞ E [Zn].

(b)   Let Wn = log2 Xn. Find E [Wn] and E rlog2(Zn/Z0)l as a function of n.

(c)   There is a constant α such that limn→∞(1/n)[log2(Zn/Z0)] = α WP1. Find α and explain how this follows from the SLLN.

(d)   Using (c), show that limn→∞ Zn = β WP1 for some β and evaluate β.

(e)   Explain carefully how the result in (a) and the result in (d) are compatible. What you should learn from this problem is that the expected value of the log of a product of IID rv s might be more significant than the expected value of the product itself.

Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.