Suppose X1, X2, X3, X4 be i.i.d. normal random variables with mean 0 and variance o, where o is the unknown parameter. Consider the following estimators: T1 X- X2 + X4, T2 (X? + X3 + X3), T3 E X7, T4 = L(X-X), and T, = |X1- X2l. %3D %3D (a) Is T unbiased for o?, for i = 1,2,3, 4? (b) Among the estimators T,...,T for o, which has the smallest MSE? (c) Is T, unbiased for o? If not, find a constant k so that kT, is unbiased for o. Evaluate the MSE of Ts.

Suppose X is a random variable with standard beta distribution and parameters (alpha) = 8 and (beta) = 2 What is (PX>=0.6)?

Suppose the time spent by a randomly selected student who uses a terminal connected to a local time-sharing computer facility has a gamma distribution with mean 12 min and variance 36 min². (a) What are the values of a and B? α = B = (b) What is the probability that a student uses the terminal for at most 18 min? (Round your answer to three decimal places.) (c) What is the probability that a student spends between 15 and 27 min using the terminal? (Round your answer to three decimal places.)

A study of blood samples of 64 men shows a mean count of 5million RBC per cubic ml., with a variance of 0.06. A tabulation of64 women from the same population group show a mean countof 4.5 million, with a variance of 0.13. What is the probability offinding a woman with a count of less than 5 million?

If V1, V2, ... , Vn1 and W1, W2, ... , Wn2 are inde-pendent random samples of sizes n1 and n2 from normal populations with the means μ1 and μ2 and the common variance σ2, find maximum likelihood estima-tors for μ1,μ2, and σ2.

5. Suppose the time spent by a randomly selected student at a campus computer laboratory has a gamma distribution with mean 20 min and variance 80 min². (a) What are the values of a and B? (b) What is the probability that a student spends between 20 and 40 min at the laboratory? (c) Only 1% of students will spend more than how many minutes at the laboratory?

Suppose a random sample of 25 students is selected from a community college where the scores on the final exam (out of 125 points) are normally distributed having mean equal to 112 and standard deviation equal to 4. Find the probability that the sample mean deviates from the population mean µ = 112 by no more than 3.

Let X1, X2, X3, ..., X, be a random sample from a distribution with known variance Var(X,) = o², and unknown mean EX, = 0. Find a (1 – a) confidence interval for 0. Assume that n is large.

no. 2

A random sample of 10 subjects have weights with a standard deviation of 10.9654 kg what is the variance of their weights be sure to include the appropriate units with the result