Question

1. Let $X_1, dots, X_n$ be a random sample from a Poisson distribution with mean $lambda$. Let the prior distribution of $lambda$ be gamma with parameters $alpha$ and $ heta$, where $alpha$ and $ heta$ are known. a) Find the posterior distribution of $lambda$ given $X_1 = x_1, dots, X_n = x_n$. b) For the squared error loss function, find the Bayesian estimate of $lambda$. c) Find the maximum likelihood estimate of $lambda$. d) Rewrite the Bayesian estimate of $lambda$ from part b) as a linear function of the MLE estimate from part c).

          1. Let $X_1, dots, X_n$ be a random sample from a Poisson distribution with mean $lambda$. Let the prior distribution of $lambda$ be gamma with parameters $alpha$ and $	heta$, where $alpha$ and $	heta$ are known.
a) Find the posterior distribution of $lambda$ given $X_1 = x_1, dots, X_n = x_n$.
b) For the squared error loss function, find the Bayesian estimate of $lambda$.
c) Find the maximum likelihood estimate of $lambda$.
d) Rewrite the Bayesian estimate of $lambda$ from part b) as a linear function of the MLE estimate from part c).

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1. Let X1, dots, Xn be a random sample from a Poisson distribution with mean lambda. Let the prior distribution of lambda be gamma with parameters alpha and heta, where alpha and heta are known.
a) Find the posterior distribution of lambda given X1 = x1, dots, Xn = xn.
b) For the squared error loss function, find the Bayesian estimate of lambda.
c) Find the maximum likelihood estimate of lambda.
d) Rewrite the Bayesian estimate of lambda from part b) as a linear function of the MLE estimate from part c).

Added by Zachary J.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

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Solved by Expert Adi S

03/16/2023

Step 1

To find the posterior distribution of $\lambda$, we need to use the Bayes' theorem: $$ P(\lambda | X_1, X_2, \dots, X_n) = \frac{P(X_1, X_2, \dots, X_n | \lambda) \cdot P(\lambda)}{P(X_1, X_2, \dots, X_n)} $$ Since the prior distribution of $\lambda$ is gamma Show more…

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Let X1, ..., Xn be a random sample from a Poisson distribution with mean λ. Let the prior distribution of λ be gamma with parameters α and θ, where α and θ are known. a) Find the posterior distribution of λ given X1 = x1, ..., Xn = xn. b) For the squared error loss function, find the Bayesian estimate of λ. c) Find the maximum likelihood estimate of λ. d) Rewrite the Bayesian estimate of λ from part b) as a linear function of the MLE estimate from part c).

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