Question 6 a) On 17th March 2020, Malaysia recorded its first death of COVID-19. Since then, the

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Question 6

a) On 17th March 2020, Malaysia recorded its first death of COVID-19. Since then, the number of death is increasing. Table 7 shows the probability distribution of number of deaths in a day for 22 days’ period since 17th March 2020.

Table 7: Probability distribution of number of deaths is a day

0 1 2 3 4 5 6 7 8
0.091 0.182 0.182 0.227 0.137 0.091 0.045 0.000 0.045

i. Find the probability that at most 2 deaths occur in a day.
ii. Find the probability that at least 5 deaths occur in a day.
iii. Find the probability that at least 3 deaths but not more than 5 deaths occur in a day.
iv. Based on Table 7 data for 22 days, find mean and variance of the number of deaths occur in a day.

b) According to World Health Organization (WHO), the risk of dying if infected by COVID-19 for a patient with a given pre-existing condition is shown in Table 8.

Table 8: Probability of dying for a patient with a given pre-existing condition

Pre-existing Condition Probability
Cardiovascular Disease 0.105
Diabetes 0.073
Chronis Respiratory Disease 0.063
Hypertension 0.060
Cancer 0.056
No Pre-existing Conditions 0.009

Suppose that among 56 COVID-10 patients in a hospital, there are 8 patients with cardiovascular disease, 4 patients with diabetes, 10 patients with chronic respiratory disease, 8 patients with hypertension and 5 patients with cancer.
i. Suppose that patients with cardiovascular disease are selected at random, what is the probability that at most two patients will die?
ii. Suppose that patients with diabetes are selected at random, what is the probability that at least two patients will die?
iii. Suppose that patients with chronic respiratory disease are selected at random, what is the probability that at most one patients will recover?
iv. Suppose that patients with cancer are selected at random, what is the probability that at least two but no more than four patients will recover?
v. Suppose that patients with no pre-existing conditions are selected at random, what is the probability that all patients will recover?

c) The death rate (probability of dying if infected with COVID-19) for a male patient is 0.047 while for a female patient is 0.028. Suppose that 7 male cases and 5 female cases are randomly selected.
i. Assuming independence trials, what is variable X and how X is distributed?
ii. What is the probability that the first death among female patients occurs in the 3rd cases?
iii. An experimental drug will be introduced to COVID-19 patients. What is the probability that among the infected male patients, the first patient recover after taking the drug is found at the fifth trial?

d) A team of scientists are collaborating to identify how quickly COVID-19 can spread from person to person. The team produces a serial interval of COVID-19 which defined as the time duration between a primary case (infector) developing symptoms and secondary case (infectee) developing symptoms. To obtain reliable estimates of the serial interval, they obtained data on 468 COVID-19 transmission events reported in mainland China outside of Hubei Province between 21st January 2020 and 8th February 2020. The team of scientists find that COVID-19 serial intervals better resemble a normal distribution than other more commonly distributions. They found that the distribution is having a mean of 4 days and standard deviation of 5 days.
What is the probability that the time duration between infector and infectee is
i. more than 6 days?
ii. between 2 to 4 days?
iii. less than 0 days? It means that there is a possibility of asymptomatic transmission (transmission of the virus from an infector who does not develop any symptoms)