Question

(1 point) A random variable with probability density function p(x; 0) = 0x0–1 for 0 <x< 1 with unknown parameter 0 > 0 is sampled three times, yielding the values 0.64,0.65,0.54. Find each of the following. (Write theta for 0.) (a) The likelihood function L(0) = d (b) The derivative of the log-likelihood function [ln L(O)] = dᎾ (c) The maximum likelihood estimate for O is is Ô =

Answer 1

TOPIC:Maximum likelihood estimator(MLE).

- Here, x1, x2, X3 Ix = 0.64, X2 = 0.65 = 0.54 are three random samples froom the following probability & density Junction p(x;o) ; osxsl. 0-1 a ox The likelihood function is 2(0) 2 T Þai ;o). 121 = TT Toxies 20). 3 zo 3 Txi ial 8-1 291 ai dy 3 M. X2 X3 0.64 X0.65 X 0:54 ? [0.64 x0.65 (0.22464 3 0-1

i The log-likelihood function is → In [110) – in [0% (0.22464) 0] = In () + in [(0.2464) = 3. In (0) + (0-1. In (0.22464. likelihood function i. The derivative of the log- is => [n { 21013] [3. inco) s. In (O) +(2-). In (0.22464) + im (0.22464) dó 3 3 -1.493256 of u. The likelihood equation is ~ [in {[(0)]] -0. =) 2 + In (0.22469) 0. 3 - -In (0.22464. 29 0 3 -In (0.22464) = (2.00g MLE + LÔ MLE of o. (Ane).