留学生代写 - 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.

Questions

1. [10 marks]

You must use the attached file "UK CPI.xls" to answer this question. 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(\frac{CPI_t}{CPI_{t-12}}) \times 100]. Draw a time-series plot and a histogram of annual inflation 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]

Prove that if P(B) > 0, then P(A|B) = 1-P(Ā|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 [\frac{d^2}{dq^2}q^y = y(y-1)q^{y-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

[f(y) = \begin{cases} ce^{-\frac{1}{2}y}, & y \geq 0, \ 0, & \text{elsewhere.} \end{cases}]

Answer the following questions.

a) [5 marks] If two such fuses have independent life lengths Y₁ and Y₂, find their joint cumulative distribution function of Y₁ and Y₂.

b) [5 marks] Suppose that 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 Y₁ + Y₂) are smaller than 2 hours (keep two decimal places in percentages).

10. [10 marks]

Let Y₁ and Y₂ have a joint density function given by

[f(y_1, y_2) = \begin{cases} cy_1, & 0 \leq y_2 \leq y_1 \leq 1, \ 0, & \text{elsewhere.} \end{cases}]

Answer the following questions.

a) [5 marks] Are Y₁ and Y₂ independent? Why?

b) [5 marks] Find the conditional density function of Y₁ given Y₂ = 1/2. Then based on this conditional density function, find P(Y₁ ≤ 3/4|Y₂ = 1/2)(keep two decimal places in percentages).

联系我们

WeChat：pythonyt001
Email: [email protected]