Step 1 Let's approach the calculation step by step. Explanation: Given Data: - The density function of the observations given (tau) is: \[ f(x|\tau) = (2\pi)^{-1/2}\tau^{1/2}e^{-0.5\tau x^2}, \] where > 0. - The prior distribution for is a Gamma distribution with parameters a and b: \[ \tau \sim \Gamma(a, b). \] The density of the Gamma distribution is: \[ f(\tau) = \frac{b^a}{\Gamma(a)}\tau^{a-1}e^{-b\tau}, \] with mean \( \frac{a}{b} \) and variance \( \frac{a}{b^2} \). Step 2 Task: 1. Derive the posterior distribution of given \( x_1, \dots, x_n \). 2. For observed data with n = 37 and \( \sum x_i^2 = 87.62 \), and prior with a = 4.25, b = 3.75, determine the values of A and B for the posterior distribution \( \Gamma(A, B) \). Explanation: Solution: 1. Derive the posterior distribution: - The likelihood of observing the data given is: \[ L(\tau) = \prod_{i=1}^{n} f(x_i|\tau). \] - We ignore the constant \( (2\pi)^{-n/2} \) and focus on the terms: \[ L(\tau) \propto \tau^{n/2} e^{-0.5\tau \sum x_i^2}. \] - The posterior is proportional to the likelihood times the prior: \[ P(\tau|x_1, \dots, x_n) \propto L(\tau) \times \pi(\tau), \] where \( \pi(\tau) \) is the prior. - Plugging in the prior and likelihood: \[ P(\tau|x_1, \dots, x_n) \propto \tau^{a-1+n/2} e^{-\tau(b + 0.5\sum x_i^2)}. \] 2. Calculating A and B: - In the posterior distribution \( \Gamma(A, B) \), the shape parameter