wo random variables, x and y, with values 0, 1, or 2, have the joint probability mass fun n in the table below. x=0 x=1 x=2 y=0 0.23 0.08 0.05 y=1 0.15 0.13 y=2 0.11 0.07 0.04
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- Consider a real random variable X with zero mean and variance σ2X . Suppose that wecannot directly observe X, but instead we can observe Yt := X + Wt, t ∈ [0, T ], where T > 0 and{Wt : t ∈ R} is a WSS process with zero mean and correlation function RW , uncorrelated with X.Further suppose that we use the following linear estimator to estimate X based on {Yt : t ∈ [0, T ]}:ˆXT =Z T0h(T − θ)Yθ dθ,i.e., we pass the process {Yt} through a causal LTI filter with impulse response h and sample theoutput at time T . We wish to design h to minimize the mean-squared error of the estimate.a. Use the orthogonality principle to write down a necessary and sufficient condition for theoptimal h. (The condition involves h, T , X, {Yt : t ∈ [0, T ]}, ˆXT , etc.)b. Use part a to derive a condition involving the optimal h that has the following form: for allτ ∈ [0, T ],a =Z T0h(θ)(b + c(τ − θ)) dθ,where a and b are constants and c is some function. (You must find a, b, and c in terms ofthe information…Suppose the joint PMF of random variables X and Y isP(X = x, Y = y) =3/14 , if (x, y) = (−1, 1)2/14 , if (x, y) ∈ {(0, 1),(−1, 0),(0, 0)}1/14 , if (x, y) ∈ {(1, 1),(1, 0),(−1, −1),(0, −1),(1, −1)}0, otherwise(a) Find the covariance of X and Y .(b) Find the correlation of X and Y .Let X and Y be random variables, and a and b be constants. ???? a) Show that Cov [aX,bY] = abCov [X,Y] . b) Show that if a > 0 and b > 0, then the correlation coefficient between aX and bY is the same as the correlation coefficient between X and Y . c) Is the correlation coefficient between X and Y unaffected by changes in the units of X and Y ?
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