f(x, y) = 2 xy² 30 x = 1, 2, 3; y = 1, 2 Determine the marginal distributions of X and Y, and then determine if X and Y independent using these distributions.
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- Let X be a random variable distributed as a Normal with mean 0 and variance 2 (i.e., X ~ N(0, 2)). Let Y be a random variable distributed as a Normal with mean 2 and variance 3 (i.e.., Y ~ N(2, 3)). X and Y are independent. What is the distribution of 2X – 5Y ?Let X Y and Z be independent normal random variables with E (X) = Mx₂ E (Y) = My E (Z) = M², var (x) = 0²x Var (y) = 0²y₁ var (7) = 0²³₂ Let u =x+Z and V=Y+Z 2 2 a. What is the marginal distribution of U? b. What is the marginal distribution of V? c. Obtan Cov (UV) d. What is the disjoint distribution of (U,V)?Let 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.
- 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 w₁, W2, ..., wt be independent random variables from a normal distribution with mean 0 and variance o². Suppose find the followings. (a) E(xt). (b) Var(xt). (c) Cov(wt, Wt-1). (d) Cov(xt, xt-1). xt = W₁+ W₂ + + WtrSuppose that Y is a uniform continuous random variable on the interval (1,11). Calculate the expected value of the random variable (Y− 1)². Use one decimal place accuracy.
- Let Y, represent the th normal population with unknown mean 4, and unknown variance of for i=1,2. Consider independent random samples, Y₁, Y2. Yin, of size n₁, from the ith population with sample mean Y, and sample variance S² = ₁₁-1(Y₁-₁². j=1 (g) For non-zero constants a's, what is the distribution of U₂ = a₁Y₁-a₂Y₂? State all the relevant parameters of the distribution. (h) Find the standard error of U₂ in part (g), assuming that of = 0² = 0². (i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the two population means, #₁ and μ2, when o² = 0 = 0² is known. (j) Define appropriate rejection regions, in terms of Y₁ - Y2, for testing Ho: #₁ = 2 against a two-sided alternative hypothesis at the a level of significance.Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.X and Y be standard normal random variables with correlation p. Compute the joint and marginal distributions of X+Y and X-Y. Are X+Y and X−Y independent?