(Solved): A bus company has three stations in Khobar ( K1,K2, and K3 ) and two statio ...

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A bus company has three stations in Khobar ( $$\mathrm{K}1,\mathrm{K}2$$, and $$\mathrm{K}3$$ ) and two stations in Riyadh (R1 and R2). Assume busses carry passengers from one of the stations in Khobar to one of the stations in Riyadh only. We know that probability of a bus starting from $$\mathrm{K}1$$ is $$0.20$$, Probability of a bus starting from K 2 is $$0.65$$, and Probability of a bus starting from $$\mathrm{K}3$$ is $$0.15$$. If: The probability of a bus going to $$\mathrm{R}1$$ is $$0.60$$ given that it started from $$\mathrm{K}1$$, The probability of a bus going to $$\mathrm{R}2$$ is $$0.40$$ given that it started from $$\mathrm{K}1$$, The probability of a bus going to $$\mathrm{R}1$$ is $$1.00$$ given that it started from $$\mathrm{K}2$$, The probability of a bus going to $$\mathrm{R}1$$ is $$0.55$$ given that it started from $$\mathrm{K}3$$, The probability of a bus going to $$\mathrm{R}2$$ is $$0.45$$ given that it started from $$\mathrm{K}3$$, find the following a) Probability that a bus will go to R2, b) Probability that a bus started from $$\mathrm{K}1$$ given that it reached $$\mathrm{R}1$$, c) Probability that a bus did not start from $$\mathrm{K}3$$ given that it reached R1, d) Probability that a bus starts from $$\mathrm{K}1$$ and goes to $$\mathrm{R}2$$, e) If you do not have the schedule of busses for that bus company, which station in Khobar would you go to in order to increase the chances of finding a bus that leaves that station and goes to R2 (justify your answer). Problem 2 A 100-floor building has 5 elevators (E1, E2, E3, E4, and E5). To get to your office in the $$100^{\text{th }}$$ floor, you may take either E1 or E2, which go from floor 1 to floor 50 only. From floor 50 , you can either take E3 directly to floor 100 or take E4 to floor 75 followed by E5 from floor 75 to floor 100. Assume that different elevators do not work on a specific day independently from each other and that $$\begin{array}{l}P(\text{ E1 not working })=0.10\\ P(\text{ E2 not working })=0.20\\ P(\text{ E3 not working })=0.30\\ P(\text{ E4 not working })=0.15\\ P(\text{ E5 not working })=0.25\end{array}$$ a) What is the probability that on a specific day, you will NOT be able to reach your office because of elevator problems. b) If you are sick on any day with probability of $$0.05$$, what is the probability that on a specific day you will not go to office because of elevator problems and being sick. A random variable $$\mathrm{X}$$ has the pdf $$f_X(x)=?\begin{array}{ll}x& 0<x?1\\ 1/2& 1<x?2\\ 0& \text{ elsewhere }\end{array}$$ A random variable $$Y$$ is obtained by transforming $$X$$ using the transformation $$Y=(X?1{)}^2$$. Find and $$\underline{\text{ Sketch }}$$ the pdf of $$Y$$ for all values of $$??<Y<?$$. (For the sketch, you can evaluate the pdf of $$Y$$ at several points and then connect them). Problem 4 The output voltage $$\mathrm{X}$$ from a binary digital receiver is a Gaussian random variable with one of the two distributions: - If bit zero was transmitted, the received signal $$X$$ is Gaussian with $$a_x=0$$ and $${?}_x=0.2$$. - If bit one was transmitted, the received signal $$X$$ is Gaussian with $$a_x=1$$ and $${?}_x=0.2$$. The decision threshold at the receiver is set at $$0.6$$ such that: If the received signal $$X>0.6$$, a binary one is decided. If the received signal $$X?0.6$$, a binary zero is decided. Assume that a binary one was transmitted, find the probability of correct decision. Problem 5 Let $$X(t)$$ and $$Y(t)$$ be independent, wide-sense stationary (WSS) random processes with zero means and the same covariance function $$C_X(?)$$. Let $$Z(t)$$ be given by $$Z(t)=X(t)\cos ?t+Y(t)\sin ?t$$ a) Find $$C_Z(t,t+?)$$, the autocovariance function of $$Z(t)$$, b) Is $$Z(t)$$ WSS?