Assuming that Y1 and Y2 obey the binary normal distribution, with parameters μY1=μY2= 0, σ2Y1= 1, σ2Y2=2, and ρ = 1/√2. Find linear transformation x1 = a11Y1+A12Y2, x2 = A21Y1+A22Y2, Make X1 and X2 the independent standard normal and randomly change.
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Assuming that Y1 and Y2 obey the binary
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- If X is a random variable with expectation µ and variance cµ2 , where c is a constant. Find a variance stabilizing transformation of X.Suppose that Yt follows the Moving Average process of order 1 (MA(1)) model Yt=ϵt−θϵt−1, where ϵt is i.i.d. with E(ϵt)=0 and Var(ϵt)=σϵ2 . a) Compute the mean and variance of Yt b) Compute the first two autocovariances of Yt c) Compute the first two autocorrelations ofLet X1, .... Xn be a random sample from a population with location pdf f(x-Q). Show that the order statistics, T(X1, ...., Xn) = (X(1), ... X(n)) are a sufficient statistics for Q and no further reduction is possible?
- Let X1 and X2 be two independent random variables. Suppose each Xi is exponentially distributed with parameter λi. Let Y=Min (X1, X2). A) Find the pdf of Y. B) Find E(Y). Hint: Let Y = Min (X1, X2). 1. P[Y > c] = P[Min (X1, X2) > c] = P[X1 > c, X2 > c] 2. Obtain the pdf of Y by differentiating its cdf of Y.Find the variance by calculating the first two moments of the random variable X = (- 1 / λ) ln (1-U), where U ~ U (0,1) and λ> 0.1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.
- 1. Suppose that Yt follows the Moving Average process of order 1 (MA(1)) model Yt=ϵt−θϵt−1, where ϵt is i.i.d. with E(ϵt)=0 and Var(ϵt)=σϵ2 . a) Compute the mean and variance of Yt b) Compute the first two autocovariances of Yt c) Compute the first two autocorrelations of YtConsider 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…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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