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P a y x y x p a y a x a xy d are x and y

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p A y x y x p A Y A X A XY , , , , , , d) Are X and Y conditionally independent, given that they belong to group A? Why/why not? e) Find and plot ( 29 y x p Y X . f) Compute and plot E(X|Y ). Use this to find E(X). g) Find ( 29 ( 29 ( 29 ( 29 ( 29 Y E Y X E X E - - . (This expectation is known as the covariance of X and Y, cov(X,Y)) h) If cov(X,Y)=0 then X and Y are said to be uncorrelated. Are X and Y uncorrelated? i) Some other joint PMFs (all uniform) are given. For each case, state whether X and Y are independent, uncorrelated? 3) After collecting some data, the rms voltage, V, at the mains is modeled as a Gaussian (although rms values are never negative) with mean 220 V and variance 100 V 2 . Answer the following questions using “the standard normal table” on page 155 of the textbook. a) Find the probability that V < 200. b) Find the probability that 205 < V < 235. c) Find the probability that V > 250. 4) a) Find and plot the cumulative distribution function of an exponential random variable whose mean value is 3. b) Using the CDF you obtained in part-(a) find the probability to have an observation in [1,3]. 5) X is an exponential random variable with mean 4. a) Find and plot ( 29 ( 29 A x F A x f A X A X , where the event A is { } a X A = , a is a positive constant b) Find [ ] A X E . How are [ ] A X E and [ ] X E related to each other? -4 -3 -2 -1 0 1 2 3 4 5 6 7 k p X ( k ) 1/12 x y 1 3 2 4 x y 3 2 x y 1 3 2 4 3 x y 2 -2
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