Question

4. Wireless Communications: Background: Per recitation, a wireless communication signal traveling from a cellphone tower to your phone bounces off numerous obstacles, causing multiple copies of the transmitted signal with different delays to arrive at your phone. These copies can add constructively or destructively, resulting in an effect called "multipath fading". The simplest (and perhaps most common) model for such is Rayleigh fading, which is a consequence of the (celebrated) Central Limit Theorem that we will learn later in the course Problem: Suppose that a we are concerned with the random power gain X caused by multipath fading between the cellphone tower and your phone. Under a Rayleigh fading assumption, 1-e-t,t0 0 t<0. (a) Suppose the voice on your cell phone only sounds good if X >0.4. Find P(X > 0.4) (b) Suppose the phone works for data if X > 1.4. Find the probability that the phone works for voice but not data; that is, find P(0.4< X <1.4) (c) [True story, simplified] When I consulted for a company a number of years ago, their phones had to work for voice 95% of the time. You should have found in part (a) that a required power threshold of 0.4 is not good enough to achieve 95% reliability. We need a better design to lower that threshold. . To find the signal power threshold at which the company's phones had to work, find to such that P(X > to) 0.95 . Recalling that this is a power threshold, how many dB less power does this system require to work than that given in part (a)? 5. Consider an experiment with sample space S and events A,B,C, and D with the following probabil ities: P(AUB) = 5. P(A) = . P(CnD)-, P(C)-. Furthermore. A and B are mutually exclusive (ie. An -φ), while C and D are independent (ie. P(cn D-P(C)P(D). Note: I know this looks like a lot of parts, but these are all short, quick answers!) (a) Find P(AnB) (b) Find P(B) (c) Find P(AnBe) (d) Find P(AUB) (e) Are A and B independent? (f) Find P(D) (g) Find P(CUD) (h) Find P(CD) (i) Find P(CnDe) G) Find P(CUD) (k) Find P(CnDe) (I) Are C and D independent?

Answer 1