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a. Let X be a single observation from a normal distribution with mean and with variance 02, where > 0. Find the maximum likelihood estimator of 0².

a. Let X be a single observation from a normal distribution with mean and with variance 02, where > 0. Find the maximum likelihood estimator of 0².

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a. Let X be a single observation from a normal distribution with mean and with variance 02, where > 0. Find the maximum likelihood estimator of 0². b. Let X₁, X2, . . . , Xn be independent identically distributed random variables from a N(μ, o2) distribution where the variance o² is known. We want to test Ho : μ = μo against H, : μ + μο (i) Derive the likelihood ratio test. (ii) Let A be the likelihood ratio. Show that -2log A is a function of (X - μo). Assuming that Ho is true, find P(-2 log > 3.84).
Transcribed Image Text:a. Let X be a single observation from a normal distribution with mean and with variance 02, where > 0. Find the maximum likelihood estimator of 0². b. Let X₁, X2, . . . , Xn be independent identically distributed random variables from a N(μ, o2) distribution where the variance o² is known. We want to test Ho : μ = μo against H, : μ + μο (i) Derive the likelihood ratio test. (ii) Let A be the likelihood ratio. Show that -2log A is a function of (X - μo). Assuming that Ho is true, find P(-2 log > 3.84).
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