LI Mathematical Methods for Statistics and

Econometrics

Latex Assignment

Important Notes

Please read the following carefully.

• The assignment consists of 10 questions. Each question is worth 10 marks.

• You must submit on canvas a pdf file generated by Latex. There is NO need

to submit the tex file. All the answers (like equations, graphs, tables, text, etc.)

must be well presented and clearly readable.

• Full marks will ONLY be awarded by showing and explaining all the steps in

every derivation. Make sure to mention which identities, theorems, proposition,

lemmas, etc... you are using in your answers.

• Partial credits will be given by showing how you have attempted to solve the

questions even if the final answer is wrong.

• No marks will be given to any copy of hand-writing attached in the submission.

• The assignment due date is on canvas.

1

1. [10 marks] You must use the attached file ”UK CPI.xls” to answer this ques-

tion. The ”UK CPI.xls” contains the UK CPI series from 1988 to October 2024,

downloaded from ONS. The annual inflation rate of the month t (in %) is defined

as ln( CPIt )×100. Draw a time-series plot and a histogram of annual inflation

CPIt−12

for the sample period January 2010 to October 2024. Then briefly comment on

your drawings. (Hint: you can use Excel to create both figures. The figures must

be readable and self-illustrated, and you must refer to the relevant figure in your

text. You also need to consider the layout of your figures.)

¯

2. [10 marks]ProvethatifP(B) > 0, thenP(A|B) = 1−P(A|B). (Hint: following

the proof of Theorem 2.7)

3. [10 marks] There are 52 cards in a standard deck with Ace to King in 4 different

suits (Hearts, Diamonds, Clubs, Spades). If we draw 5 cards randomly, which of

the following is more likely to happen?

a) 4 cards of the same value

b) 5 cards in sequential order (i.e., Ace, 2,3,4,5 or 10,J,Q,K,Ace), but not in

the same suit

c) 5 cards in the same suit, but not in sequential order

You must show how you calculate the probabilities and list the exact answers in

a table (keep four decimal places in percentages).

4. [10 marks] Suppose that currently 1 in 5000 people in the UK is affected by

COVID-19. A diagnostic test for COVID-19 has a sensitivity of 95% (i.e., if a

person has COVID-19, the test correctly identifies it 95% of the time) and a false

positive rate of 2% (i.e., if a person doesn’t have COVID-19, the test incorrectly

indicates a positive result 2% of the time).

a) [5 marks] If a randomly selected person takes the test and the result is

positive, what is the probability that the person actually has COVID-19

(keep two decimal places in percentage)?

b) [5 marks] What if the false positive rate is 0.2%. What do your results

imply?

5. [10 marks] Let Y have a Poisson distribution with mean λ. Find E[Y(Y −1)]

and then show that Var(Y) = λ.

6. [10 marks] Let Y be a random variable following a geometric distribution with

p, where p is the probability of success. Then, what is E[Y(Y − 1)] equal to?

(Hint: use the fact that d2 qy = y(y −1)qy−2)

dq2

2

7. [10 marks] Suppose that Y has probability density function f(y) = c(y − 1)

between 1 and 3, and f(y) = 0 elsewhere. Compute the value of Var(Y) and

Var(Y2) (keep two decimal places).

8. [10 marks] Let the random variable Y measure the length of a Zoom meeting

in hours. It is known that E(Y) = 2. What is the probability that the meeting

takes NO more than 2 hours (keep two decimal places in percentages)?

a) [5 marks] Assume that Y follows an exponential distribution.

b) [5 marks] Assume that Y follows a normal distribution.

9. [10 marks] The length of life, denoted as Y, for a type of fuse has a probability

density function

(cid:40)

ce−1y, y ≥ 0,

2

f(y) =

0, elsewhere.

Answer the following questions.

a) [5 marks] If two such fuses have independent life lengths Y and Y , find

1 2

their joint cumulative distribution function of Y and Y .

1 2

b) [5 marks] Suppose that the these two fuses are used in a system where one

fuse is in the primary system and the other is in a backup system that only

comes into use if the primary system fails. Find the probability that the

total life length of the two fuses in this system (denoted as Y + Y ) are

1 2

smaller than 2 hours (keep two decimal places in percentages).

10. [10 marks] Let Y and Y have a joint density function given by

1 2

(cid:40)

cy , 0 ≤ y ≤ y ≤ 1,

1 2 1

f(y ,y ) =

1 2

0, elsewhere.

Answer the following questions.

a) [5 marks] Are Y and Y independent? Why?

1 2

b) [5 marks] Find the conditional density function of Y given Y = 1/2. Then

1 2

based on this conditional density function, find P(Y ≤ 3/4|Y = 1/2)(keep

1 2

two decimal places in percentages).

3