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Suppose that we have data y = (v1.....yn). Each data-point y, is assumed to be generated by a distribution with the following probability density function: Pyin) knye, y ≥ 0. The unknown parameter is ŋ, with x assumed to be known, and 77, K > 0. (a) Write down the likelihood for n given y. Find an expression for the maximum likelihood estimate (MLE) ). (b) A Gamma(a, 3) distribution is chosen as the prior distribution for n. Show that the posterior distribution is also a gamma distribution with parameters that you should determine.

Suppose that we have data y = (v1.....yn). Each data-point y, is assumed to be generated by a distribution with the following probability density function: Pyin) knye, y ≥ 0. The unknown parameter is ŋ, with x assumed to be known, and 77, K > 0. (a) Write down the likelihood for n given y. Find an expression for the maximum likelihood estimate (MLE) ). (b) A Gamma(a, 3) distribution is chosen as the prior distribution for n. Show that the posterior distribution is also a gamma distribution with parameters that you should determine.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.3: Special Probability Density Functions
Problem 2E
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Suppose that we have data y = (y1,..., yn). Each data-point y; is assumed to be generated by a distribution with the following probability density function: P(v₁|n) = kny¹e, y₁ ≥ 0. The unknown parameter is ŋ, with x assumed to be known, and ŋ,<> 0. (a) Write down the likelihood for n given y. Find an expression for the maximum likelihood estimate (MLE) . (b) A Gamma (a,ß) distribution is chosen as the prior distribution for n. Show that the posterior distribution is also a gamma distribution with parameters that you should determine.
Transcribed Image Text:Suppose that we have data y = (y1,..., yn). Each data-point y; is assumed to be generated by a distribution with the following probability density function: P(v₁|n) = kny¹e, y₁ ≥ 0. The unknown parameter is ŋ, with x assumed to be known, and ŋ,<> 0. (a) Write down the likelihood for n given y. Find an expression for the maximum likelihood estimate (MLE) . (b) A Gamma (a,ß) distribution is chosen as the prior distribution for n. Show that the posterior distribution is also a gamma distribution with parameters that you should determine.
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