3. Find the sampling distribution of Y for the following population distributions: iid (a) Y₁,..., Ynd Poisson (A) (find instead the distribution of nỸ); iid (b) Y1,..., Yn exponential(0); ~ iid (c) Y₁,..., Yn ~ Bernoulli (p) (find instead the distribution of nỸ).
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- 3. Find the sampling distribution of Y for the following population distributions: (а) Yi,..., Yn iid Poisson(A); iid (b) Y1,..., Yn exponential (0); iid (c) Y1,..., Yn ´Bernoulli(p) (find instead the distribution of nỸ).Let Y, represent the ith normal population with unknown mean 44, and unknown variance of for i 1,2. Consider independent random samples, Ya, Y₁2,,Yin, of size ni, from the ith population with sample mean Y, and sample variance S?=1Σj=1(Y₁j - Y₁². (a) What is the distribution of Y;? State all the relevant parameters of the distribution. (b) Find a level a test (that is, the rejection region) for testing Ho : μi = μio versus Ha: Pipio when of is unknown and n, is small. (c) In the context of the test in part (b), state the Type I error and give a probability statement for the level of significance, a.= 16. A simple random sample of size n = 64 is obtained from a population with u = 77 and o = (a) Describe the sampling distribution of x. (b) What is P (x>79.5) ? (c) What is P (xs72.9)? (d) What is P (75.9 79.5) = (Round to four decimal places as needed.) (c) P (xs72.9) = (Round to four decimal places as needed.) (d) P (75.9Q1: Calculate the mean and the variance for Weibull Distribution when (i) a = 4,ẞ= 1 (ii) α = 1,ẞ = 4 Q2: Calculate the mean and the variance for Exponential Distribution when (i) a = 4,ẞ = 1 (ii) a = 1,ẞ= 4 Q3: Calculate the mean and the variance for Beta Distribution when (i) a = 4,ẞ= 1 (ii) a = 1,ẞ = 4In the daily production of a certain kind of rope, the number of defects per foot given by Y is assumed to have a Poisson distribution with mean ? = 4. The profit per foot when the rope is sold is given by X, where X = 70 − 3Y − Y2. Find the expected profit per foot.Let Y, represent the ith normal population with unknown mean #, and unknown variance of for i=1,2. Consider independent random samples, Ya,Ya. Yin, of size n,, from the ith population with sample mean Y, and sample variance S?=₁1(Yu - Y.². (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 = o2 = 0². (i) Discuss how the distribution of Y₁ - ₂ can be used to test the equality of the two population means, #, and p2, when of=o=o² is known. (j) Define appropriate rejection regions, in terms of Y₁ - ₂, for testing Ho: #₁ = ₂ against a two-sided alternative hypothesis at the a level of significance.