Use r to answer the following

5. (a) State the Spectral Theorem for Symmetric matrices. (b) Write out the spectral factorization of 5 2 A = 2 2 (c) Using the change of variables x = Qy, where Q is an appropriate orthogonal matric, express the quadratic form 5r, + 4r142 + 21% in terms of y and y2. Describe and sketch the graph of the equation 5r, + 4712 + 2x2 = 1 in the T1, 12 plane. (d) Find the spectral decomposition of A.-12/02 3. [5 points] Suppose that the number of customers arriving at the Woori Bank follows a Poisson process with a rate of Aw = 0.5 per minute, the number of customers arriving at the Hana Bank follows a Poisson process with a rate of An = 0.2 per minute, and the two Poisson processes are independent. Let( We )be the waiting time to have the kth customer at the Woori Bank and /, be the waiting time to have the kth customer at the Hana Bank. Find the probability distribution of the random variable Wa H 2 and explain why.l. a) (Poisson superposition) Flies and wasps land on your dinner plate in the manner of inde- pendent Poisson processes with respective intensities 5L and pt. Show that the arrivals of ying objects form a Poisson process with intensity A + pt. b) (Poisson thinning) Insects land in the soup in the manner of a Poisson process with intensity 1, and each such insect is green with probability p, independently of the colours of all other insects. Show that the arrivals of green insects form a Poisson process with intensity 11p. 7. (12 points) Consider a birth and death process with birth rate i + 1 and death rate i when there are i individuals. (a) Starting at state 2, find the expected time until the process first leaves state 2. (b) Find the expected time to go from state 2 to state 4