Problem 1. (a) Generate 200 replicas of uniform [−π,π] and 200 normal with mean 0 and standard deviation 1/8. Set data x from this uniform, error ε from this normal distribution. The response y is by model:
y = sin(x)+ε
Fit the data with two types of smoothing techniques. Plot both the data and your fitted smooth curves. (b) The same as (a) except changing the standard deviation from 1/8 to 1/2.
[Remark: Use a computer for you calculation; explain your analysis and results carefully]
Problem 2. (a) Use a linear regression model to analyze the GAG in urine data in data frame GAGurine. Produce a chart to help a pediatrician to assess if a child’s GAG concentration is ‘nor- mal’ or not (hint: plot in one graph the estimated line and confidence bands at different levels)
(b) Consider using a smooth regression to analyze the GAG in urine data
[Remark: See the data set named “GAGurine.csv” in the assignment. Use a computer for you calculation; explain your analysis and results carefully]
Problem 3. Service times of a queuing system follow Exponential distribution with an unknown pa- rameter θ. A sample of service times X1, X2, · · · , Xn is observed.
(a) Show that the Gamma(α, λ) family of prior distributions is conjugate.
(b) Find the posterior parameters, posterior mean and variance. (As functions of θ,α,λ and Xi’s)
(c) Suppose that we can allow α = 0 and consider a prior density π(λ) = 1/λ for λ > 0. Of course, this is not a proper density. Nevertheless, find the posterior distribution, its mean and variance.
[Remark: Solve this problem by paper and pencil. Write down the detailed calculation process.] Problem 4. Write a computing code to calculate the integration of 5 (x3 − x2)e−x2/2 using Monte
−5
Carlo simulation with N samples from a uniform distribution, for N = 10, 100, 1000. For each choice
of N, repeat the experiment for 500 times, compute the variance and visualize the relationship between the variance and N.
[Remark: Use a computer for you calculation; explain your results carefully]
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