Question

Consider the probability density function f(x) = (3θ + 1) x3θ,    0  ≤  x  ≤  1....

Consider the probability density function f(x) = (3θ + 1) x3θ,    0  ≤  x  ≤  1.

The random sample is 0.859, 0.008, 0.976, 0.136, 0.864, 0.449, 0.249, 0.764. The moment estimator of θ based on a random sample of size n is .055. Please answer the following:

a) find the maximum likelihood estimator of θ based on a random sample of size n. Then use your result to find the maximum likelihood estimate of θ based on the given random sample.

b) find the maximum likelihood estimate of the median.

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