7.70. Let X,...,x,; Y,., Y,; Z,..,Z, be respective independent random samples from three normal distributions N(u,a+...
Consider independent random samples of size n, and n, from respective normal distributions, Xi ~ Nuh, σ ) and Y, ~ Num σ ). 30 (a) Derive the GLR test of Ho : σ|-σ1 against H. σ. σ1, assuming that and μ2 are known. (b) Rework (a) assuming that andHa are unknown. Consider independent random samples of size n, and n, from respective normal distributions, Xi ~ Nuh, σ ) and Y, ~ Num σ ). 30 (a) Derive the...
Let independent random samples, each of size n, be taken from the k normal distributions with means u cd [j - (k 1)/2], j = 1, 2,..., k, respectively, and common variance o2. Find the maximum likelihood estimators of c and d
Let the independent normal random variables Y1,Y2, . . . ,Yn have the respective distributions N(μ, γ 2x2i ), i = 1, 2, . . . , n, where x1, x2, . . . , xn are known but not all the same and no one of which is equal to zero. Find the maximum likelihood estimators for μ and γ 2.
Suppose that X and Y are independent random variables with the same unknown mean u. Both X and Y have a variance of 36. Let T = aX + bY be an estimator of u. What condition must a and b satisfy in order that T be an unbiased estimator for ? Is T a normal random variable?
8.7-11. Let Y1,Y2, ...,Yn be n independent random variables with normal distributions N(Bx;,02), where X],x2,...,xn are known and not all equal and B and 2 are unknown parameters (a) Find the likelihood ratio test for Ho: B = 0 against H: B+0. (b) Can this test be based on a statistic with a well-known distribution?
(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y ~ N(5,9) (we use the notation N (?, ?. ) ). Let W 3X-2Y + 1. (a) Compute E(W) and Var(W) (b) It is known that the sum of independent normal distributions is n Compute P(W 6)
5. Suppose that X and Y are independent with distributions N(0,0) and N(0,02), respectively. Let Z=X+Y. Also, let W = 02X – oʻY. Prove that Z and W are uncorrelated.
Question 5 Consider two normal populations N(H) and N(u2, 1). Let X, and 12 be the sample means of random samples from these two populations, respectively 1) (1 point) Find a pivotal quantity for Δ-μ,-,42, and derive the l-a confidence interval based on this pivotal quantity. 2) (1 point) State the relationship between the test and the confidence interval in 1). Question 5 Consider two normal populations N(H) and N(u2, 1). Let X, and 12 be the sample means of...
Suppose X, Y and Z are three different random variables. Let X obey Bernoulli Distribution. The probability distribution function is p(x) = Let Y obeys the standard Normal (Gaussian) distribution, which can be written as Y ∼ N(0, 1). X and Y are independent. Meanwhile, let Z = XY . (a) What is the Expectation (mean value) of X? (b) Are Y and Z independent? (Just clarify, do not need to prove) (c) Show that Z is also a standard...
12 marks Let independent random samples of sizes n and n2 be taken respectively from two normal distributions with unknown means 1 and 2 and unknown variances oand o. Denote the two samples by . . ,Jn, and y,... , yn2: Which have means T and T, and sample variances s and s2, respectively (a) 4 marks Show that when of = o2, the likelihood ratio test statistic for testing Ho 12 against H 2 can be written as T2...