1. Bayesian Statistics - Maxwell. Please explain answer too.

Sampley1, . . . , yn, comes from Maxwell distribution with a density f(y|?)=sqrt(2/pi) theta^(3/2) y^2 e^(-theta(y^2)/2)

Assume an exponential prior on theta, ?(?) =?e^(???) for ? >0, ? >0.

(a) Show that posterior belongs to gamma family and depends on data via summation(y_i^2)

(b) For?= 1/2 andy_1= 1.4, y_2= 3.1, andy_3= 2.5, find Bayes estimator for?. How does the Bayes estimator compares to the MLE and prior mean. The MLE for?is?3n/summation((y_i)^2)

(c) Using Python to calculate the 95% equitailed credible set for?.

(d) Find a prediction for a future single observation. For this, you will need the mean ofMaxwell, which isE[Y] = 2sqrt(2/??).

1. Maxwell. Sample y1, ..., Un, comes from Maxwell distribution with a density f(y|0) = 1 N 03 / 2 2 2 e - 042 / 2, y > 0, 0 > 0. Assume an exponential prior on 0, TT (0) = de -, 0 > 0, > > 0. (a) Show that posterior belongs to gamma family and depends on data via Et-1 y, (b) For A = 1/2 and y1 = 1.4, y2 = 3.1, and y3 = 2.5, find Bayes estimator for 0. How the Bayes estimator compares to the MLE and prior mean. The MLE for 0 is 3n (c) Using MATLAB/Octave/R/Python to calculate the 95% equitailed credible set for (d) Find a prediction for a future single observation. For this, you will need the mean of Maxwell, which is E[Y] = 2 2