Question

Let $X$ be a binomial random variable with parameters $n$ and $\Theta$. Suppose that $\Theta$ is uniform in the interval $[0,1]$. Consider the following cost function which emphasizes the errors at the extreme values of $\theta$ : $$ C(g(X), \theta)=\frac{(\theta-g(X))^2}{\theta(1-\theta)} $$ Show that the Bayes estimator is given by $$ g(k)=\frac{\Gamma(n)}{\Gamma(k) \Gamma(n-k)} \frac{k}{n} . $$

   Let $X$ be a binomial random variable with parameters $n$ and $\Theta$. Suppose that $\Theta$ is uniform in the interval $[0,1]$. Consider the following cost function which emphasizes the errors at the extreme values of $\theta$ :

$$
C(g(X), \theta)=\frac{(\theta-g(X))^2}{\theta(1-\theta)}
$$
Show that the Bayes estimator is given by

$$
g(k)=\frac{\Gamma(n)}{\Gamma(k) \Gamma(n-k)} \frac{k}{n} .
$$

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Probability, Statistics, and Random Processes For Electrical Engineering

Probability, Statistics, and Random Processes For Electrical Engineering

Alberto Leon-Garcia 3rd Edition

Chapter 8, Problem 93

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Step 1

We need to find the Bayes estimator \( g(k) \) for the binomial random variable \( X \) with parameters \( n \) and \( \Theta \), where \( \Theta \) is uniformly distributed over the interval \([0, 1]\). The cost function given is \( C(g(X), \theta) = Show more…

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Let $X$ be a binomial random variable with parameters $n$ and $\Theta$. Suppose that $\Theta$ is uniform in the interval $[0,1]$. Consider the following cost function which emphasizes the errors at the extreme values of $\theta$ : $$ C(g(X), \theta)=\frac{(\theta-g(X))^2}{\theta(1-\theta)} $$ Show that the Bayes estimator is given by $$ g(k)=\frac{\Gamma(n)}{\Gamma(k) \Gamma(n-k)} \frac{k}{n} . $$

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