1.36. A professor has two light bulbs in his garage. When both are burned out. they are replaced, and the next day starts with two working light bulbs. Suppose that when both are working, one of the two will go out with probability 0.02 (each has probability 0.01 and we ignore the possibility of losing two on the same day). However, when only one is there, it will burn out with probability 0.05. (i) What is the long-11m fraction of time that there is exactly one bulb working? (ii) What is the expected time between light bulb replacements? PRoBLEM 10.4 A chemical solution contains N molecules of type A and an equal number of molecules of type B. A reversible reaction occurs between type A and B molecules in which they bond to form a new compound AB. Suppose that in any small time interval of length h, any particular unbonded A molecule will react with any particular unbonded B molecule with probability och + O[h), where or is a reaction rate of formation. Suppose also that in any small time interval of length hI any particular AB molecule disassociates into its A and B constituents with probability ll + DUI), where fl is a reaction rate of dissolution. Let X (t) denote the number of AB molecules at time t. Model X(t} as a birth and death process by specifying the parameters. Note that one mole of molecules is N = 6.02214129 X 1023. Find the mean and Auto Correlation function of X (t) = Acos (cut + $) when w and are fixed. and A is a random variable with Normal density function of zero mean and Variance equal to two. Is X(#) a wide sense stationary random process