Question

(1 point) Suppose f(x) = wxW-1,0 0 is a density function for a continuous random variable X. Find the maximum likelihood estimator for w. (a) Which of the following is the likelihood function L(w) = [] f(x;)? i=1 OA. L(w) = w"(x1 + x2 + ... + xnw-1 OB. L(w) = nw(xi · X2 ... Xn)-! OC. L(w) = nw(x1 + x2 + ... + xn)w-1 OD. L(w) = w"(x· X2 --- x)(w=1) OE. L(w) = w"(x1 · X2 . Xn)w-! OF. none of the above (b) Which of the following is the maximum likelihood estimator for w? O In(x1 • X2 ... Xn) A. W = n 1 B. û n In(x1 + x2 + ... + xn) -n c. W = In x In x2 + ... + In xn -n D. W = In(xi : x2 * x) O E. none of the above (C) Find the maximum likelihood estimate for w based on the data below: 0.6, 0.88, 0.17.0.41,0.75, 0.15 0.25 w =

Answer 1

• (a) The likelihood function is represented by L(w) = nw(x1 * x2 * ... * xn)^w-1.
• (b) The maximum likelihood estimator for w can be calculated by taking the derivative of the logarithm of the likelihood function and setting it to zero to find the maximum value. The maximum likelihood estimator for w is w = n / (x1 * x2 * ... * xn).
• (c) To find the maximum likelihood estimate for w based on the provided data (0.6, 0.88, 0.17, 0.41, 0.75, 0.15, 0.25), we can substitute these values into the formula for the maximum likelihood estimator: w = n / (x1 * x2 * ... * xn).

Given the data, w = 7 / (0.6 * 0.88 * 0.17 * 0.41 * 0.75 * 0.15 * 0.25). Performing the calculation will provide the maximum likelihood estimate for w.

Step-by-step explanation:

(a) The likelihood function L(w) is a representation of the probability of observing the given data points based on the parameter w in the density function. It is expressed as nw(x1 * x2 * ... * xn)^w-1, where n is the number of data points.

(b) To find the maximum likelihood estimator for w, we aim to maximize the likelihood function. By taking the logarithm of the likelihood function and differentiating it with respect to w, we find the value of w that maximizes the likelihood, which results in w = n / (x1 * x2 * ... * xn).

(c) Applying the formula for the maximum likelihood estimator to the provided dataset yields the specific value of w based on the observed data points. Substituting these values into the formula allows for the calculation of the maximum likelihood estimate for w, offering insight into the parameter that best fits the given data according to the specified distribution.