6. Find the variance of the F distribution for v2> 4, by making use of the...
2. If X and Y are independent random variables, X has a normal distribution with mean 2 variance 4, and Y has a chi-square distribution with 9 degrees of freedom, then find u such that P(X > 2+11,7)=0.01.
4. A sample of size n-81 is taken from an exponential distribution with the pdf f(x)-Be-6x, θ > 0, x > 0. The sample mean is i-35. Find a 95% large- sample confidence interval for θ using the Central Limit Theorem.
4. Let 8 >0. Let X, X2,..., X, be a random sample from the distribution with probability density function S(*;ð) - ma t?e-vor x>0, zero otherwise. Recall: W=vX has Gamma( a -6, 0-ta) distribution. Y=ZVX; = Z W; has a Gamma ( a =6n, = ta) distribution. i=1 E(Xk) - I( 2k+6) 120 ok k>-3. 42 S. A method of moments estimator of 8 is 42.n 8 = h) Suggest a confidence interval for 8 with (1 - 0) 100%...
4. Find the Fisher Information and the Cramer-Rao lower bound for the variance of an unbiased estimator of θ given a random sample . , xn from the density r3 -z/θ where x > 0 and f(x:0-6 94e θ > 0.
5. We wish to conduct a hypothesis test of the form Ho : μ1-2-0 vs Ha : μί-μ2 > 0. Both populations are assumed normal with equal variance and samples are assumed independent. We draw 15 observations for the first sample and 17 observations for the second sample. State the distribution of the test statistic for this test. (a) t-distribution with 32 degrees of freedom (b) t-distribution with 30 degrees of freedom (e) Normal distribution with 30 degrees of freedom...
Problem 4 (20 points). Show how to use the chi-square distribution to calculate P(a < S2 < b), where S,: nii Σί! from N(μ, σ2) (Xi - X)2 is the sample variance of a random sample Xi,.. . ,Xn
Let h be an exponentially-distributed random variable with the distribution function p- exp(-x) for x > 0 and ph = nction Ph 0 for a s 0. Derive the distribution function of its square root, Solution: 2y exp(-y2
QUESTION 5 The statistical results of the test in Question 4 are as follows: F (2, 147)-1.24, p>.05. (1) In the above results, degrees of freedom within or df2 has what value? (2) What is the value of the test statistic? (3) What is the decision about the Null Hypothesis?
5. Use Rice's Theorem to prove the undecidablity of the following language. P = {< M > M is a TM and 1011 E L(M)}.
7 The publisher of a sports magazine plans to offer new subscribers one of three gifts: a sweatshirt with the logo of their favorite team, a coffee cup with the logo of their favorite team, or a pair of earrings also with the logo of their favorite team. In a sample of 431 new subscribers, the number selecting each gift is reported below. At the 0.02 significance level, is there a preference for the gifts or should we conclude that...