# 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 $$H_1: \theta \leq 0.80$$