= Let X₁, ..., X, be a random sample from EXP(0), and define 0₁ Đ, = nX/(n + 1). 2 (a) Find the variances of ₁ and ₂. (b) Find the MSEs of ĝ₁ and ĝ₂. CENT (c) Compare the variances of , and 8₂ for n = 2. X and
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- IfX1 andX2 are the means of independent random samples of sizes n1 and n2 from a normal population with the mean μ and the variance σ ^2 , show that the variance of the unbiased estimator ω·Xbar1+(1−ω)·Xbar2is a minimum whenω=n1/(n1+n2)A snack food manufacturer estimates that the variance of the number of grams of carbohydrates in servings of its tortilla chips is 1.25. A dietician is asked to test this claim and finds that a random sample of 22 servings has a variance of 1.35. At ⍶ = 0.05 is there enough evidence to reject the manufacturer’s claim?A snack food manufacturer estimates that the variance of the number of grams of carbohydrates in servings of its tortilla chips is 1.34. A dietician is asked to test this claim and finds that a random sample of 16 servings has a variance of 1.22. At α=0.05, is there enough evidence to reject the manufacturer's claim? Assume the population is normally distributed. Complete parts (a) through (e) below.
- A snack food manufacturer estimates that the variance of the number of grams of carbohydrates in servings of its tortilla chips is 1.141.14. A dietician is asked to test this claim and finds that a random sample of 2424 servings has a variance of 0.950.95. At alpha equals 0.10α=0.10, is there enough evidence to reject the manufacturer's claim? Assume the population is normally distributed.Let the random variable X be defined on the support set (1,2) with pdf fX(x) = (4/15)x3, Find the variance of X.If X1, X2, and X3 constitute a random sample of sizen = 3 from a normal population with the mean μ and thevariance σ2, find the efficiency of X1 + 2X2 + X34relative toX1 + X2 + X33as estimates of μ.
- If X is a uniformly distributed random varibale with a=9 and b=16, then Calculate the variance of X? Round to three decimal placesSuppose X1, ..., Xn have been randomly sampled from a normal distribution with mean 0 and unknown variance sigma^2, and let U = c * i=1 -> n summation (X_ i)^2 , where c is a constant. Find the value of c that minimises the Mean Squared Error (MSE)A finance analyzer have come to the conclusion that he wants to purchase stocks from Company A, with an expected yield of E(X) = 4% and variance to the yield being V(X) = 0.49. Another stock from another Company B have an expected yield of E(Y) = 6% and variance V(Y) = 0.64. The correlation of the yield between the two stocks is p(X,Y) = 0.3. The finance analyzer wants to invest a portion p(0 < p < 1) into Company A stocks, and the rest (1 - p) into company B stocks. The combined investment have a yield of W = pX + ( 1 - p)Y. For p = 0.4, such that W = 0.4X + 0.6Y. What is the expected value and variance for W.
- A discrete random variable X has the following probability distribution: Value of X 1 2 3 4 5 6 Probability 0.30 0.20 0.15 0.15 0.10 0.10 Find the probability that X is a multiple of 2. ["", "", "", ""] Find the probability X is greater than 1, or P(X > 1). ["", "", "", ""] Find the mean of the random variable X, μX. ["", "", "", ""] Find the variance of the random variable X, σ2XσX2 . ["", "", "", ""] Find the standard deviation of the random variable X, σX.Suppose X, Y, Z are iid observations from a Poisson distribution with parameter λ, which is unknown. Consider the 3 estimators T1 = X + Y − Z, T2 = 2X + Y + Z 4 , T3 = 3X + Y + Z 5 . (a) Which among the above estimators are unbiased? (b) Among the class of unbiased estimators, which has the minimum variance?