8 . [15 points] A camera has a detector array of N pixels: Each time a photon arrives at the camera. it strikes exactly one of the N pixels. chosen uniformly at random. The lo :ations are independent from one photon to the next. In this problem. we are interested in answering the following question: What is the ex- pected number of photons that must arrive before all N of the pixels have been struck? You should provide the exact answer for the cases N = 4 and N = 1m}. (Hint: W'hen N = 2. it takes 3 photons. on average. to ensure that both of the pixels have been struck.) You may answer this question either using random Matlab simulations or by deriving an analytic solution and using a calculatorfllatlab when necessary. Your nal answers should be provided in decimal form and should be accurate to the nearest integer to receive full credit. (Hint: if you choose to deriv r the answer analytically. you might let X1 be the number of photons required to strike a pixel {clearly X1 = 1). let X2 be the number of additional photons required to strike a second unique pixel after one pixel has been struck {X2 will be random]. let X9. be the number of additional photons required to strike a third unique pixel after two unique pixels have been struck. etc. Then letting X denote the total. E[X] = E[X1+X2+- - -+X_.-.,r].]