Statistics MATH 324 Assignment 1 complete solutions correct answers key

Statistics MATH 324 Assignment 1 complete solutions correct answers key

McGill University

Q1 (a) Show that the mean and variance of a Poisson random variable X with parameter

_ > 0 satisfy E(X) = var(X) = _. Hint: Calculate EfX(1 􀀀 X)g.

(b) Compute the mean and the variance of a Gamma random variable X with parameters

_; _ > 0. Hint: Recall the definition of the Gamma function 􀀀 and that

for any _ > 0, 􀀀(_ + 1) = _􀀀(_).

(c) Calculate the moment generating function M of the Geometric distribution with

parameter p 2 (0; 1). For which t 2 R does M(t) exist? Use M to compute the

mean and variance of the Geometric distribution.

Q2 Which distribution would be a suitable model for the random phenomenon described

below? Justify your choice. Identify which parameters are unknown and which are

given from the context.

(a) Number of cases of E.coli infections in the state of Colorado in a given year.

(b) Number of fish, out of 20, who survive 24 hours in a water tank filled with

contaminated water.

(c) Total number of bags filled with coffee beans that need to be examined until 10

bags with spoiled beans are found.

(d) The actual measured resistance of a wire manufactured for use in a computer

system.

(e) Size of an insurance claim due to a car accident.

(f) Relative humidity measured at a specific location.

Q3 A bottling machine can be regulated so that it discharges an average of _ ounces

per bottle. It has been observed that the amount of fill dispensed by the machine is

normally distributed with _ = 2:0 ounces.

(a) If n = 9 bottles are randomly selected from the output of the machine, what is

the probability that the sample mean differs from the true mean _ by at most

0.3 ounces?

(b) Find the probability that the sample mean differs from the true mean _ by at

most 0.3 ounces using sample sizes n = 25, n = 36 and n = 64. What pattern

do you observe among the values of this probability that you computed for the

various sample sizes? Can you provide an explanation for your finding?

(c) How does the probability obtained in part (a) change when _ is unknown, and

the sample variance equals S2n

= 5:3?

Q4 A forester studying the effects of fertilization on certain pine trees is interested in

estimating the average basal area of pine trees.

(a) From many years of experience, he finds that the measurements of basal areas are

approximately normally distributed with standard deviation of about 4 square

Johanna G. Nešlehová

Statistics MATH 324

McGill University, Fall Term 2015

Assignment 1: Due October 1 at 11:59 PM

inches. Suppose the forester would like the sample mean basal area to be within

1 square inch of the population mean, with probability 0:90. How many trees

must he measure in order to ensure this degree of accuracy?

(b) Suppose you are not convinced about the accuracy of the forester’s guess of the

population standard deviation and prefer to regard this quantity as unknown. If

a random sample of n = 9 basal areas is to be measured, find two statistics g1

and g2 such that

Pr(g1 _ ( _X

n 􀀀 _) _ g2) = 0:90:

Q5 Let S2n

denote the sample variance for a random sample of n = 10 ln(LC50) values for

copper and let S2m

denote the sample variance for a random sample of m = 8 ln(LC50)

values for lead, both samples using the same species of fish. The population variance

for measurements on copper is assumed to be twice the corresponding population

variance for measurements of lead. Assume S2n

to be independent of S2m

.

(a) Using an appropriate statistical table, find numbers a and b such that

Pr

_

S2n

S2m

_ b

_

= 0:95; Pr

_

S2n

S2m

_ a

_

= 0:95:

Hint: Notice that Pr(U1=U2 _ k) = Pr(U2=U1 _ 1=k).

(b) If a and b are as in part (a), calculate

Pr

_

a _

S2n

S2m

_ b

_

:

Q6 In questions Q3 and Q4, the discharges per bottle and the basal area measurements

were assumed to be normally distributed. Suppose now this assumption is dropped.

(a) Revisit questions Q3 (a) and (b) and Q4 (a). Do your previous calculations still

make sense? Justify your answers and interpret the results (i.e., are they exact

or approximate, how good you think the approximation is)?

(b) Redo questions Q3 (c) and Q4 (b), assuming the probabilities to be calculated

are only approximate. Compare your findings with your answers to Q3 (c) and

Q4 (b).

Johanna G. Nešlehová

Statistics MATH 324

McGill University, Fall Term 2015

Assignment 1: Due October 1 at 11:59 PM

Q7 (optional) Suppose that X1; : : : ;Xn is a random sample from some distribution F such that

E(X1)4 < 1. Using the Weak Law of Large Numbers, prove that the sample variance

converges in probability as n ! 1 to the true variance of the underlying distribution.

You can use, without proof, the following two results about convergence in probability:

(1) Suppose that Y1; Y2; : : : and Z1;Z2; : : : are sequences of random variables such

that Yn

P!

a and Zn

P!

b as n ! 1 for some a; b 2 R. Then

Xn + Yn

P!

a + b and XnYn

P!

ab as n ! 1:

(2) Suppose that Y1; Y2; : : : is a sequence of random variables such that Yn

P!

a as

n ! 1 for some a 2 R. Let further g be an arbitrary continuous function. Then

g(Yn) P!

g(a) as n ! 1:

Q8 (optional) The following R function returns the sample variances of N samples of size n drawn

from the Normal distribution with mean _ = 0 and variance _2 = 1:

sample.var <- function(n=10,N=100){

data <- rnorm(n*N,mean=0,sd=1)

data.mat <- matrix(data,ncol=N)

apply(data.mat,2,var)

}

For example, to calculate sample variances of 1; 000 samples of size 8, call

sample.var(n=8,N=1000)

(a) Draw a histogram of sample variances corresponding to 1; 000 samples of size

5 from the Normal distribution with mean _ = 0 and variance _2 = 0. What

is the mean of the sample variances? Is it close to _2? What is the sample

variance of the sample of sample variances? Add a plot of the theoretical density

of the sampling distribution of the sample variance. Does the theoretical density

provide a good approximation to the histogram? Careful: In class, we calculated

the distribution of (n 􀀀 1)S2n

=_2, not S2n

.

(b) Repeat part (a) when the sample size is n = 500. How do the results compare?

What does the theoretical density of the sample variance remind you of?

(c) Adapt the function sample.var so that it draws samples from the Exponential

distribution with parameter 1. Redo parts (a) and (b) when the random samples

are drawn from the Exponential distribution with parameter 1. Although not

justified theoretically, you can still overlay the histograms by the theoretical

sampling density of the sample variance of a Normal sample. What do you see

now?