Let Y₁, Y₂,.. Yn denote a random sample from a uniform distribution with density function 1, 0 ≤ y ≤0, f(y|0) = 0, elsewhere. = For testing Ho: 000 vs. Ha: 000, show that the a-level Likelihood Ratio Test (LRT) gives RR={Y(n) > 00} U{Y(n) < 00α=}.
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- let X and Y be independent standard normal random variables. Determine the pdf of W = X^2 + Y^2. Find the mean and the variance of U= (1/2)WLet Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s^2is known.Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix using GLMConsider a random variable Y with PDF Pr(Y=k)=pq^(k-1),k=1,2,3,4,5....compute for E(2Y)
- calculate the variance of g(X) =X^2+2, where X is the random variable with probability distribution : x=0,1,2,3; f(x) = 0.1, 0.2, 0.4,0.3 . a. 19.65 b. 26.28 c. 10.65 d. 11.65 e. 14.65Find the least square estimate for beta (the slope) given that its y-intercept is 0. Y_i = BetaX_i + e_i where e_i are independent and identically distributed N(0, variance_e) randon variables i = 1,...,2.Let the probability function of the random variable X be defined as f(x)=c/x^4 for x>1, otherwise f(x)=0. Calculate the constant c, the expected value and the variance.
- Show that the variance of a geometric random variable X; is given by V ar(X) = q/p^2 .Let X and Y be independent random variables, each with mean=mu and variance=sigma^2. Find the expression of the correlation of X and XY in terms of mu and sigma.The useful life of a concrete built on the seashore has a Weibull distribution of parameter scale δ> 0 year and power β = 2. (δ=21). Determine a) The average life of concrete EXb) The variance VXc) The probability that the concrete lasts more than 10 years; P (X> 10)
- Prove that for the binomial population with density function f (x, p) ="C, p`q" - ', x = 1 , 2,...n. 21. the maximum likelihood estiator for p is . Find its variance.If independent random variables 'X' and 'Y' have the variances 36 and 16 respectively, find the correlation coefficient between (X + Y) and (X- Y) assuming X and Y are zero mean.Let Y₁, Y2,..., Y₁ be a random sample whose probability density function is given by -fammet, 0, f(y:B)=684 200 (=1 0 0 elsewhere and suppose that n = 200, y = 20, y = 100, 200, i) Derive the standard error of ß, se(B) = 0.0009, using MLE approach. ii) Find an approximate 95% Confidence interval for B. y = 250 and 3 = 0.025.