Homework No.3 (MSDS 954:567) solution

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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 ‘normal’ 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 parameter 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 R 5
−5
(x
3 − x
2
)e
−x
2/2 using Monte
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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