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- Suppose that the rate of increase of a bighorn ram's horn is approximated by 0.42 x² - 3.97 x + 31.24 cm per year, where x is between 2 and 5 years. (These numbers have been randomized from the actual equation given in the project done in discussion section.) Find the total change in the length of a bighorn ram's horn between the ages of 2 and 5. Give the answer as a decimal number with at least two decimal places.Let X be a strictly positive random variable. Determine which of the following quantitiesare larger:(a) E(X^4) or (E(X))^4(b) E(X^1/9) or (E(X))^1/9(c) E(X^3 + 4X^2 + 9) or (E(X))^3 + 4(E(X))^2 + 9(d) E(e^X ) or e^E(X)(e) E(arctan(X)) or arctan(E(X))Draw all possible samples each of size 2 from the population 2, 4, 6 and 8 using sampling with replacement. Find mean of each sample and verify that (6) Mean of = u and (ii) V(X) (ii) V(R) %3D
- Prove the followingQ Let Z be a standard normal random variable 2. Show that E[Z^*'] = n E [z"-¹]. to compute E [24] Use the resultSuppose model (XY, XZ, YZ) holds in a 2 x 2 x 2 table, and the common XY conditional log odds ratio at the two levels of Z is positive If the XY and YZ conditional log odds ratios are both positive or both negative, show that the XY marginal odds ratio is larger than the XY conditional odds ratio.
- Let X₁, X2, X3, Xn be a random sample with unknown mean EX; = µ, and unknown variance Var(X₂) = o². Suppose that we would like to estimate 0 = μ². We define the estimator as 2 • - (™)² - [ 2x]* Xk to estimate 0. Is an unbiased estimator of ? Why?B) Let X1,X2, .,Xn be a random sample from a N(u, o2) population with both parameters unknown. Consider the two estimators S2 and ô? for o? where S2 is the sample variance, i.e. s2 =E,(X, – X)² and ở² = 'E".,(X1 – X)². [X = =E-, X, is the sample mean]. %3D n-1 Li%3D1 [Hint: a2 (п-1)52 -~x~-1 which has mean (n-1) and variance 2(n-1)] i) Show that S2 is unbiased for o2. Find variance of S2. ii) Find the bias of 62 and the variance of ô2. iii) Show that Mean Square Error (MSE) of ô2 is smaller than MSE of S?. iv) Show that both S2 and ô? are consistent estimators for o?.t x, x, ... x be a random sample from an exponential distribution with parameter 0. Find sufficient estimator for '8'.