Suppose X₁,..., Xn are independent exp(X) random variables, where X is the rate parameter. Find UMVUE for λ.
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- Calculate E{X}, E{Y }, Var{X}, Var{Y } and ρ XY for random variables X and Y with jointdensity functionLet Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s2 is known. Find the Score function, the estimating equation and the information matrix.Consider two independent exponential random variables X1 and X2 with parameter lambda=1. LetY1 = X1 Y2 = X1 + X2. Find the MMSE estimate of Y1 using Y2.
- Let X and Y be independent standard normal random variables. Determine the pdf of W = x² + y². Find the mean and the variance of U = /W. Let Y₁, Y₂, ..., Yn denote a random sample of size n from a population with a uniform distribution on the interval (0, 0). Consider = Y(1) = min(Y₁, Y₂, ..., Y₁) as an estimator for 0. Show that is a biased estimator for 0. Let X and Y be independent exponentially distributed random variables with parameter X λ = 1. If U = X + Y and V =. Find and identify the marginal density of U. X+YLet X and Y be two random variables with joint probability mass function: p(x,y) = 1/48 xy(1+y) ??? x=1,2,3 ??? y=1,2 p(x,y) = 0, Otherwise. Please enter the answer to 2 decimal places. What is the variance if (4-2X)?Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.
- Let X1, X2,..Xn be a random sample from a Poisson distribution with variance 0. Find the UMVUE for Pe(X=0). R(Ctrl) -Let X, Y be independent random variables with exponential distribution of parameter θ > 0. Are the random variables Z = X + Y and W = X / (X+Y) independent?Let x be random variable with E(x) = 2 and var(x) = 3. Verify that random variable x and the random variable y=-4x+8 are orthogonal. CS Scanned with CamScanner
- Suppose Y is a Poisson random variable with rate A. Let X be a random variadie be such that given Y, X is a Gamma random variable with rate 1 and shape parameter Y. Compute E(X) and Var(X).Let X1, X2,...Xn be a random sample from a Poisson distribution with variance 0. Find an unbiased estimator for Pe(X=0).