Statistical Computing

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Statistical Computing

Homework 1: RNG and Writing R functions Due Date: March 10th, 2004

1. Read p37 to 41 in R-intro.pdf about random number generator and various plots on examining the distribution of a set of data.

2. Read p45 to 48 in R-intro.pdf about “Grouping, loops and conditional execution”

and writing your own function.

3. Write a R program (or other programming language) to implement a uniform random number generator using a multiplicative congruential method with x_i+1 = 17x_i mod m and m = 2¹³ − 1. Generate 500 numbers for the starting point x₀ = 100. (help For the sequence u_i = x_i/m:

(a) Plot a histogram to display the results. (help(hist))

(b) Calculate the coefficient correlation of the pairs of successive number u_i and u_i+1. (help(cor)).

(c) Plot the pairs (ui, ui+1) on a 2D plot. (help(plot))

4. Write a program to find the period of a random number generator for a given seed. Use this program to find the period of the sequence generated by xi+1= 7xi

mod 13 and x₀ = 19. Find the period if a is changed to 3.

5. Read p69-70 and 78-79 in R-intro.pdf about how to add a graph in terms of

“Low-level plotting commands” to existing plot which is produced by “High- level plotting commands.”

6. Write a program to sample k values from the probability mass function p_i = i/55, i = 1, 2, . . . , 10. Plot the histogram of generated values for k = 50, 500 and 5000.

(One plot! Not three plots!)

7. Let a discrete random variable X has a probability mass function pj = P (X = j).

Define a new function

λ_n = P (X = n | X > n − 1) = p_n 1 −^Pⁿ⁻¹_j=1p_j.

The quantities λ_n, n ≥ 1 are called discrete hazard rates since if we think of X as the lifetime of some item then λ_n represents the probability that an item has reached the age n will die before n + 1. If we are given the mass function p, we can simulate X. Write a program to simulate X when only λ_ns are given.

For the mass function p_j = (0.9)^j, first compute the hazard function λ_n for n = 1, 2, . . . , 100. Then, use these λ_n’s in your program to simulate 500 values of X. Plot a histogram of these simulated values.

8. Derive and implement a method to generate samples of a Weibull random variable whose probability distribution function is given by

F (x) = 1 − exp(−αx^β), 0 < x < ∞.

Run your program to simulate 1000 values of Weibull random variable with α = 1 and β = 0.5.

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9. Write a program that uses the rejection method to sample a random variable having the distribution function:

F (x) =

Z ∞ 0

x^yexp(−y)dy, 0 ≤ x ≤ 1.

10. Suppose it is easy to generate a random variable from any of the distributions F_i, i = 1, 2, . . . , k. How can we generate a random variable from the distribution:

F (x) =

k

Y

i=1

Fi(x).

Hint: If X_is are random variables with distributions F_is, respectively, then what random variable X has the distribution F ?

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