Exercise 1
Consider the following model,
- Prior: \(\theta \sim \text{Beta}(\alpha = 5, \beta = 5)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)
and observed data with statistics,
- Sample size: \(n = 20\)
- Number of “successes” \(\sum x_i = 15\)
Use the given prior and the observed data to calculate a Bayes’ estimate of \(\theta\). (Use the posterior mean.)
Exercise 2
Consider the following model,
- Prior: \(\theta \sim \text{Beta}(\alpha = 50, \beta = 20)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)
and observed data with statistics,
- Sample size: \(n = 40\)
- Number of “successes” \(\sum x_i = 32\)
Use the given prior and the observed data to calculate a Bayes’ estimate of \(\theta\). (Use the posterior mean.)
Exercise 3
Consider the following model,
- Prior: \(\theta \sim \text{Beta}(\alpha = 10, \beta = 60)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)
and observed data with statistics,
- Sample size: \(n = 5\)
- Number of “successes” \(\sum x_i = 2\)
Use the given prior and the observed data to calculate a Bayes’ estimate of \(\theta\). (Use the posterior mean.)
Exercise 4
Consider the following model,
- Prior: \(\theta \sim \text{Beta}(\alpha = 10, \beta = 60)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)
and observed data with statistics,
- Sample size: \(n = 5\)
- Number of “successes” \(\sum x_i = 2\)
Use the given prior and the observed data to calculate a 99% credible interval
Exercise 5
Consider the following model,
- Prior: \(\theta \sim \text{Beta}(\alpha = 4, \beta = 4)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)
and observed data with statistics,
- Sample size: \(n = 5\)
- Number of “successes” \(\sum x_i = 2\)
Use the given prior and the observed data to calculate a 90% credible interval
Exercise 6
Consider the following model,
- Prior: \(\theta \sim \text{Beta}(\alpha = 10, \beta = 4)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)
and observed data with statistics,
- Sample size: \(n = 50\)
- Number of “successes” \(\sum x_i = 2\)
Use the given prior and the observed data to calculate a 95% credible interval
Exercise 7
Consider the following model,
- Prior: \(\theta \sim \text{Beta}(\alpha = 10, \beta = 4)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)
and observed data with statistics,
- Sample size: \(n = 50\)
- Number of “successes” \(\sum x_i = 20\)
Use the given prior and the observed data to test \(H_0: \theta > 0.50\) vs \(H_1: \theta \leq 0.50\)
Exercise 8
Consider the following model,
- Prior: \(\theta \sim \text{Beta}(\alpha = 10, \beta = 10)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)
and observed data with statistics,
- Sample size: \(n = 20\)
- Number of “successes” \(\sum x_i = 5\)
Use the given prior and the observed data to test \(H_0: 0.25 < \theta < 0.50\) vs \(H_1: \theta \leq 0.25, \theta \geq 0.50\)
Exercise 9
Consider the following model,
- Prior: \(\theta \sim \text{Beta}(\alpha = 3, \beta = 3)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)
and observed data with statistics,
- Sample size: \(n = 100\)
- Number of “successes” \(\sum x_i = 70\)
Use the given prior and the observed data to test \(H_0: \theta > 0.80\) vs \(H_1: \theta \leq 0.80\)