Question

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X, be a random sample from the distribution with the probability density function Let A0 and let X, X2, f(x; A) 24xe, x>0. a. Find E(X), where k> -8. Enter a formula below. Use* for multiplication, for divison, ^ for power, lam for A, Gamma for the r function, and pi for the mathematical constant . For example, lam k*Gamma(k/2)/pi means Akr(k/2)/T Ax2 or u =x2. Hint 1: Consider u -e"du Hint 2: I'(a) a 0 E(Xk) Submit Answer Tries 0/10 b. Obtain the method of moments estimator of A, i. Enter a formula below. Use m1 for the sample mean X. For example, 23*pi^2/m1 means 23T2/X. Your expression should NOT contain any function. Hint 3: Г(а) %3 (а— 1)Г(а — 1) Hint 4: I = x Tries 0/10 Submit Answer c. Suppose n-5, and x1-2.29, x2 2.57, x3-3.00, x4-2.56, x=2.23. Find the method of moments estimator of A. Give your answer to 4 decimal places Submit Answer Tries 0/5 d. Obtain the maximum likelihood estimator of A, A. Enter a formula below. Use m2 for the second moment of X, i.e. m2 ΣΧ Submit Answer Tries 0/10 e. Suppose n 5, and x1-2..29, x2 2.57, x3-3.00, x4 2.56, x=2.23. Find the maximum likelihood estimator of A. Give your answer to 4 decimal places

Answer 1

2) f(0) -xx2 7 427812? xxo 7+k x2 Elak) Let x² = t. 27adx = dt. -t at Elxk) = 10 x 6tk 2x3 to 6 Jo

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