Question

40. Let $X_1, \dots, X_n$ be a sample from a Bernoulli distribution with unknown parameter $p \in [0, 1]$. We want to give a Bayesian estimate for the variance $var_p(X_i) = p(1 - p)$ with respect to a $beta(\alpha, \beta)$ prior distribution with parameter $p$. (i) Determine the posterior density for $p$. (ii) Determine the Bayes estimators for $p$ and $var_p(X_i)$.

          40. Let $X_1, \dots, X_n$ be a sample from a Bernoulli distribution with unknown parameter $p \in [0, 1]$.
We want to give a Bayesian estimate for the variance $var_p(X_i) = p(1 - p)$ with respect to a
$beta(\alpha, \beta)$ prior distribution with parameter $p$.
(i) Determine the posterior density for $p$.
(ii) Determine the Bayes estimators for $p$ and $var_p(X_i)$.

40. Let X1, …, Xn be a sample from a Bernoulli distribution with unknown parameter p ∈ [0, 1].
We want to give a Bayesian estimate for the variance varp(Xi) = p(1 - p) with respect to a
beta(α, β) prior distribution with parameter p.
(i) Determine the posterior density for p.
(ii) Determine the Bayes estimators for p and varp(Xi).

Added by Joel T.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Madhur L

05/02/2023

Step 1

.., Xₙ. Since X₁,..., Xₙ are independent and identically distributed (i.i.d.) Bernoulli random variables with parameter p, the likelihood function is given by: L(p) = p^k * (1-p)^(n-k) where k is the number of successes (number of Xᵢ = 1) and n-k is the number Show more…

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40. Let X₁,..., Xₙ be a sample from a Bernoulli distribution with an unknown parameter p ∈ [0, 1]. We want to give a Bayesian estimate for the variance var(X) = p(1 - p) with respect to a beta(α, β) prior distribution with parameter p. (i) Determine the posterior density for p. (ii) Determine the Bayes estimators for p and varp(X).