关键词 > STAT2203/7203

STAT2203/7203 (S2-2023): Assignment 03

发布时间：2023-10-25

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

STAT2203/7203 (S2-2023): Assignment 03

Due: 27-October-2023 @16:59

1. [5 marks each]

We have seen how to simulate from a distribution using the inverse-transform method; see §5.8 of the course notes as well as slide 8/14 of Lecture4-3. Another method to simulate random variables from a given distribution is using rejection sampling. This question concerns a particular application of rejection sampling.

Benford’s law is a distribution on the integers {1, 2, . . . , 9} with probability mass function

fD (d) = log10 , d ∈ {1, 2, . . . , 9}.

We would like to be able to simulate the random variable D from this distribution. Suppose X has probability mass function

fX (x) =

9 ,

x ∈ {1, 2, . . . , 9}

Conditional on X = x, the random variable Y has a Bernoulli(fD (x)/ log10 (2)) distribu- tion.

(a) Verify that fD (x)/ log10 (2) ≤ 1 for all x ∈ {1, 2, . . . , 9}.

(b) What is the joint probability mass function of (X, Y)?

(d) Determine the conditional probability mass function of X given Y = 1.

(e) This suggests we can simulate a random variable with probability mass function fD using the following algorithm

Y = 0

While (Y = 0) {

Simulate X from a uniform distribution on {1,2,...,9}

Simulate Y from a Bernoulli distribution with success

probability f(X)/log10(2)

}

Return X

In each loop a new pair of random variables (X, Y) is simulated, independent of all previously simulated random variables. Implement this algorithm in R (or any programming language of your choice). You will need to use a while loop. In R, the general form is

while (cond) {

expressions

}

where cond is a length one logical vector.

(f) What is the distribution of the number of pairs of random variables (X, Y) that

need to be simulated to simulate a single random variable from Benford’s law?

2. [5 marks each]

For The following questions, work out your answers ‘by hand’. You may still use R (or any other programming language) to obtain probabilities and quantiles from the appro- priate distributions and calculate your ﬁnal answers.

A study investigated if psychotherapy combined with limited administration of Methylene- dioxymethamphetamine (MDMA) can reduce symptoms of post-traumatic stress disor- der. Severity of symptoms was measured via the CAPS-IV score with higher scores indicating more severe symptoms. Forty-eight patients recruited to the study with twenty-four patients being randomly allocated each of the two dosage levels (Low – 40 mg, High – 125 mg). The primary outcome was the reduction in CAPS-IV score one month after the end of treatment.

The forty-eight patients at the commencement of the study had an average CAPS-IV score of 81.35 with a sample standard deviation of 17.54. At the end of treatment, the High dose group experienced an average drop in CAPS-IV score of 24.2 with a sample standard deviation of 23.1. The Low dose group experienced an average drop in CAPS- IV score of 12.7 with a sample standard deviation of 19.4.

(a) Determine a 95% conﬁdence