Question

Let X1,...,Xn be mutually independent random variables, each with pdf f(x|?) = 2?x exp(-?x^2), for x > 0. What is the log-likelihood of ?? (a) 2? ? xi exp(-?xi^2) (b) n log(2?) + ? log(xi) - ? ? xi^2 (c) 2^n ?^n ? xi exp(-? ? xi^2) (d) n log(2?) + ? log(xi) - ? ? xi^2 (e) n log(2?) + ? log(xi) - ? ? xi^2 Continuing with the example in Q1, the MLE of ? is (a) (? xi^2 + ? log(xi)) / n (b) n / (? xi^2 + ? log(xi)) (c) ? xi^2 / n (d) n / (? - 2 ? xi) (e) n / ? xi^2

          Let X1,...,Xn be mutually independent random variables, each with pdf
f(x|?) = 2?x exp(-?x^2), for x > 0.
What is the log-likelihood of ??
(a) 2? ? xi exp(-?xi^2)
(b) n log(2?) + ? log(xi) - ? ? xi^2
(c) 2^n ?^n ? xi exp(-? ? xi^2)
(d) n log(2?) + ? log(xi) - ? ? xi^2
(e) n log(2?) + ? log(xi) - ? ? xi^2
Continuing with the example in Q1, the MLE of ? is
(a) (? xi^2 + ? log(xi)) / n
(b) n / (? xi^2 + ? log(xi))
(c) ? xi^2 / n
(d) n / (? - 2 ? xi)
(e) n / ? xi^2

Let X1,...,Xn be mutually independent random variables, each with pdf
f(x|?) = 2?x exp(-?x^2), for x > 0.
What is the log-likelihood of ??
(a) 2? ? xi exp(-?xi^2)
(b) n log(2?) + ? log(xi) - ? ? xi^2
(c) 2^n ?^n ? xi exp(-? ? xi^2)
(d) n log(2?) + ? log(xi) - ? ? xi^2
(e) n log(2?) + ? log(xi) - ? ? xi^2
Continuing with the example in Q1, the MLE of ? is
(a) (? xi^2 + ? log(xi)) / n
(b) n / (? xi^2 + ? log(xi))
(c) ? xi^2 / n
(d) n / (? - 2 ? xi)
(e) n / ? xi^2

Added by Alex H.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Shubham Sharma

Step 1

First, we need to find the likelihood function L(θ) for the given pdf. Since the random variables X1, ..., Xn are mutually independent, the likelihood function is the product of their individual pdfs: L(θ) = ∏[20xi exp(-θx^2)] for i = 1 to n Show more…

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Let X1,...,Xn be mutually independent random variables, each with pdf f(x|θ) = 2θx exp(-θx^2), for x > 0. What is the log-likelihood of θ? (a) 2θ ∏ xi exp(-θxi^2) (b) n log(2θ) + ∑ log(xi) - θ ∑ xi^2 (c) 2^n θ^n ∏ xi exp(-θ ∑ xi^2) (d) n log(2θ) + ∏ log(xi) - θ ∑ xi^2 (e) n log(2θ) + ∑ log(xi) - θ ∏ xi^2 Continuing with the example in Q1, the MLE of θ is (a) ∑ xi^2 + ∑ log(xi) / n (b) n / (∑ xi^2 + ∑ log(xi)) (c) ∑ xi^2 / n (d) n / (θ - 2 ∑ xi) (e) n / ∑ xi^2