AM41PB Probabilistic Modelling

Hello,

Please find the attached dataset, excel file to solve the first assignment question which is related to pareto distribution.

Please find the attached assignment and its details below. Need a report in word format with step by step derivation of the assignment( solve numerically, no programming needed except for plotting the graphs).

For the first question pertaining to pareto distribution please use the data which is 2012-13 year from the excel file.

1a) Show that the Pareto distribution is indeed a probability distribution.

Answer: Please derive and show that pareto distribution is a probability distribution.( with step by step solution and with explanation.

1b) Plot the data as the percentage of the population (y-axis) with income above certain values (x-axis); to simplify the presentation, present the income in thousands of pounds.

Answer: From the excel file please take the 2012-13 data and plot the graph.(can use any programming language to plot the graph, title for the axises must be shown.)

1c) Assuming the data is i.i.d. and represented by the Pareto distribution, find the maximum likelihood expression for the parameter λML. Obtain the numerical value of λML given data D and xm = £15, 000. Specify the data you are working with and the number of data points used..

 Answer: Derive and find the numerical value for pareto maximum likelihood.

1d) For data set D = {x1, . . . , xN }, show that the Gamma probability distribution function p(λ|α, β) = ( β α Γ(α) λ α−1 e −βλ for λ > 0, 0 for λ ≤ 0. is the conjugate prior for the Pareto distribution. Derive the corresponding posterior p(λ|D).

Need toFo prove that gamma distribution is conjugate prior for the pareto distribution and also should derive the posterior p(λ|D) detailed steps and explanation needed.

1e) Assuming the data is represented by the Pareto distribution, find the Maximum A Posteriori expression of the parameter λMAP. Use the prior values α = 1 and β = 1 to obtain the numerical value of λMAP given data D.

Derive the Maximum A Posteriori expression of the parameter λMAP. and get the numerical value for same( step by step solution and explanation)

1f) Use the values obtained for the parameters λML and λMAP to compare the estimated percentage of the population with income above £20, 000

From the answers obtained from the 1c and 1e) we need to estimate the percentage of the population with income £20, 000.

Second question decision theory:

2a) Write an expression for the misclassification error of Diamonds as White Sapphire p(D|WS) and of White Sapphire as Diamonds p(WS|D) as functions of the threshold weight gT

Need to write the expression for the misclassification error with respect to diamond and white sapphire.(step by step solution and explanation)

2b) Minimize the total misclassification error with respect to the threshold weight gT . What is the expression obtained for gT (you can use a simplified notation)?

Need the expression for minimum total misclassification error(step by step solution and explanation)

2c) Calculate the value of gT for the parameters λD = 1, µS = 5, σ2 S = 1 and prior probabilities p(WS) = 0.8 and p(D) = 0.2. Sketch the two distributions, point to the threshold value gT and to the area that contributes to the misclassification error of Diamonds as White Sapphire.

Need the numerical value and a graph showing the two classes and threshold value( step by step solution and explanation).

2d) The mine is loosing money and the owner realises that he has not taken into account that large diamonds are exponentially more expensive and the misclassification cost p(WS|D) with respect to p(D|WS) (which for simplicity we will attribute cost 1 to) is CostWS|D = e λC g . Assume λC < λD. Find the threshold weight gT expression in this case (not numerical value).

Find the GT value(derive it) numerical value not needed.

Third question which is based on probability

3a) Plot the directed acyclic graphical model that represents the problem .

Need the directed graph with nodes and edges

3b) Write an expression for the joint probability of all variables in the problem as well as the probabilities given in the question

 All the joint probabilities that are in the question and which are not in the question.

3c) What is the probability that Ruth has the disease given that James has a sore throat and elevated fever P(R = 1|S = 1, F = 1)

Need the probablity a numerical value and a step by step solution with explanantion

3d) After a while, James drinks some water and realises that his sore throat was a result of not drinking enough so that S = 0 . What is the probability that Ruth has the disease now?

Need the probablity a numerical value and a step by step solution with explanantion

3e) Later in the day James is again unclear whether he has a sore throat or not. A lateral flow test (T) shows that Ruth is positive for the disease (T = 1). However, these tests are