A sample of 4 observations (X1 = 0.4, X2 = 0.6, X3 = 0.7, X4 = 0.9) is collected from a continuous distribution with pdf
fx=θxθ-1, 0<x<1
(a) Find the point estimate of θ by the Method of Moments.
(b) Find the point estimate of θ by the Method of Maximum Likelihood. Use two decimal places.
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 =...
Let X1,X2,X3,X4 be observations of a random sample of n-4 from the exponential distribution having mean 5, What is the mgf of Y-X1 X2 X3 X4? 4. 5. What is the distribution of Y? What is the mgf of the sample mean X = X+X+Xa+X1 ? 6. 7. What is the distribution of the sample mean?
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,
X3, and X4 be a random
sample of observations from a population with mean μ and
variance σ2. The observations are independent because
they were randomly drawn. Consider the following two point
estimators of the population mean μ:
1 = 0.10 X1 + 0.40
X2 + 0.40 X3 + 0.10
X4 and
2 = 0.20 X1 + 0.30
X2 + 0.30 X3 + 0.20
X4
Which of the following statements is true?
HINT: Use the definition of...
7.Let X1, X2, X3, and X4 be a random sample of observations from a population with mean μ and variance σ2. Consider the following estimator of μ:⊝1 = 0.15 X1 + 0.35 X2 + 0.20 X3 + 0.30 X4. Is this a biased estimator for the mean? What is the variance of the estimator? Can you find a more efficient estimator?
Let X1, X2,
X3, and X4 be a random
sample of observations from a population with mean μ and
variance σ2. Consider the following estimator of
μ: 1 = 0.15 X1 +
0.35 X2 + 0.20 X3 + 0.30
X4. Using the linear combination of random
variables rule and the fact that X1, ...,
X4are independently drawn from the population, calculate
the variance of 1?
A.
0.55 σ2
B.
0.275 σ2
C.
0.125 σ2
D.
0.20 σ2
(1 point) Solve the system x +x2 x2 +x3 X1 +X4 X1 X2 X3 X4 +s
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 X1, X2, X3, X4 be a random sample from a standard normal population. What is the probability distribution (give the name of the distribution and the value of any parameter(s)) of (a). (X1 - Xbar)^2 + (X2 - Xbar)^2 + (X3 - Xbar)^2 + (X4 - Xbar)^2 (b). ((X1 + X2 + X3 + X4)^2)/4
Let X1, X2, ..., Xn be a random sample from the distribution with probability density function f(x;t) = Botha, 0 < x < 2, t> -4. a. Find the method of moments estimator of t, t . Enter a formula below. Use * for multiplication, / for division and ^ for power. Use m1 for the sample mean X. For example, 7*n^2*m1/6 means 7n27/6. ſ = * Tries 0/10 b. Suppose n=5, and x1=0.36, X2=0.96, X3=1.16, X4=1.36, X5=1.96. Find the...