2. For a sample X1,..., Xn from λ f(xlu, 1) = ( 5)1/2exp(- 21 x3 1(x...
Let X1,...............,Xn be a sample from an exponential population with parameter λ. (a) Find the maximum likelihood estimator for λ. (b) Is the estimator unbiased? (c) Is the estimator consistent?
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3e-dız?, x > 0. a. Find E(X), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for \, Gamma for the function, and pi for the mathematical constant 11. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/ I. Hint 1: Consider u = 1x2 or u = x2....
Let X1,X2,...,Xn denote a random sample from the Rayleigh distribution given by f(x) = (2x θ)e−x2 θ x > 0; 0, elsewhere with unknown parameter θ > 0. (A) Find the maximum likelihood estimator ˆ θ of θ. (B) If we observer the values x1 = 0.5, x2 = 1.3, and x3 = 1.7, find the maximum likelihood estimate of θ.
Let > 0 and let X1, X2, ..., Xn be a random sample from the distribution with the probability density function f(x; 1) = 212x3 e-tz, x > 0. a. Find E(XK), where k > -4. Enter a formula below. Use * for multiplication, / for divison, ^ for power, lam for 1, Gamma for the function, and pi for the mathematical constant i. For example, lam^k*Gamma(k/2)/pi means ik r(k/2)/n. Hint 1: Consider u = 1x2 or u = x2....
Let X1,..., Xn be a random sample from the pdf f(x:0)-82-2, 0 < θ x < oo. (a) Find the method of moments estimator of θ. (b) Find the maxinum likelihood estimator of θ
6. Suppose that X1, ..., Xn is a random sample from a population with the probability density function f(x;0), 0 E N. In this case, the esti- mator ÔLSE = arg min (X; – 6)? n DES2 i=1 is called the least square estimator of Ô. Now, suppose that X1, ..., Xn is a random sample from N(u, 1), u E R. Prove that the least square estimator of u is the same as maximum likelihood estimator of u.
Let X1, X2, ..., Xn be a random sample of size n from the distribution with probability density function f(x;) = 2xAe-de?, x > 0, 1 > 0. a. Obtain the maximum likelihood estimator of 1. Enter a formula below. Use * for multiplication, / for divison, ^ for power. Use mi for the sample mean X, m2 for the second moment and pi for the constant 1. That is, n mi =#= xi, m2 = Š X?. For example,...
Let X1, X2, ..., Xn be a random sample of size n from the distribution with probability density function f(x1) = 2 Æ e-dz?, x > 0, 1 > 0. a. Obtain the maximum likelihood estimator of 1 . Enter a formula below. Use * for multiplication, / for divison, ^ for power. Use m1 for the sample mean X, m2 for the second moment and pi for the constant n. That is, m1 = * = *Šxi, m2 =...
3. Let X1 , X2, . . . , Xn be a randon sample from the distribution with pdf f(r;0) = (1/2)e-z-8,-X < < oo,-oc < θ < oo. Find the maximum likelihood estimator of θ.
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood function, L(θ), and the log-likelihood function, `(θ). (b) Find the maximum likelihood estimator of θ, ˆθ. (c) Is ˆθ unbiased? (d) What is the distribution of X? Find the moment estimator of θ, ˜θ.