Statistics 630 – Assignment 3

Instructions: (same as those given in the first assignment)

The material covered by this assignment is primarily in Lectures 06–09 and Chapter 2 of the textbook.

1. Use R to check the accuracy of the binomial approximation to the hypergeometric distribution.

You can use the dbinom and dhyper functions to compute probabilities for the two distributions.

Compare the distributions for n = 10, M/N = 0.6 and N = 50, 100, 1000. The syntax in R for

computing the hypergeometric probability mass function is dhyper(x,M,N-M,n). Note that x can be

a vector in either function. So, for example, dbinom(0:3,10,.4) returns a vector with binomial(10,.4)

probabilities for x = 0, 1, 2, 3.

2. Chapter 2 Exercise 2.3.18. Take note of the assumptions. The parameter is λt where λ = 2/min is

the rate and t is the length of the time interval. Also, events (as in part (c)) referring to separate

time intervals are independent.

3. Suppose fV (x) = c(x + x

2

)I[1,3](x) is the pdf for some random variable V that takes values in the

interval [1, 3].

(a) What is the value of c?

(b) Find P(V ≤ 2) and P(1.8 ≤ V ≤ 2.3).

(c) Find P(V

2 ≤ 6). Hint: what values of V satisfy the event?

(d) Find the cdf for V , for all values on the real line.

4. Exercise 2.4.6(a,d).

5. Chapter 2 Exercise 2.4.19. Use an indicator function to give an expression for f(x) that is valid for

all real x.

6. Chapter 2 Exercise 2.5.3(c,d,f,g). Give reasons if you say “no”.

7. Chapter 2 Exercise 2.5.5. Use the pnorm function in R. Add

(d) Find the 40th and 77th percentiles. Use the qnorm function in R.

8. Chapter 2 Exercise 2.5.13. For part (b), also identify the pmf or pdf, whichever is appropriate.

9. Chapter 2 Exercise 2.5.21. Add

(b) Find the quantile function.

10. Consider the Gumbel cdf F(x) = exp(−e

−(x−b)/a) for all real x.

(a) Find the pdf. (Use the chain rule appropriately. Check that it is nonnegative.)

(b) Identify location and scale parameters, if any.

(c) Find the quantile function. How does it depend on the parameters?