3. A Taiwan mask manufacturer received an order worth 50 million yuan from a politically unstable country (India). Given a probability of 0.7 this manufacturer can receive payment for the 50 million yuan order. There is a probability of 0.15 that you can only receive a payment of 30 million yuan, a probability of 0.1 that you can only receive a payment of 10 million yuan, and a probability of 0.05 that you will not receive any payment. For the sake of insurance, this Taiwanese manufacturer insures an accident insurance and pays an insurance premium of 10 million yuan first. The Japanese insurance company underwriting will pay any shortfalls in the company's receivables. If the random variable X represents the amount of money spent or earned by this Taiwanese mask manufacturer finally insuring here. Please calculate the expected value E(X) of X. (10 points)

6. Alice's Shop in Oxford, UK, is one of the most famous little shops in the world. It sells Alice in Wonderland themed gifts, souvenirs and memorabilia. One of the popular items is the Tea Party paper bookmark. The average of the Tea Party paper bookmark sold in a week is 2100 units. Supposed the weekly demand is approximately normally distributed, with a standard deviation of 200 units. The manager wants to limit the probability of being out of stock of the Tea Party paper bookmark. Please help to solve his problems below. You need to show your calculations. Remember to use four decimal places. a. If the manager wants to limit the probability of being out of stock of the Tea Party paper bookmark to no more than \(5\%\) in a week, how many units of the bookmarks should he keep at the start of the week? (5 points) b. If the Shop has 2500 units on hand at the start of week, what is the probability that weekly demand will be greater than the inventory? (5 points) c. What is the probability that the Shop sells between 2000 and 2500 Tea Party paper bookmarks in a typical week? (6 points) d. If the standard deviation of weekly demand for the toy increases from 200 units to 350 units, how many more Tea Party paper bookmarks would have to be stocked to ensure that the probability of weekly demand exceeding inventory is no more than \(10\%\)? (5 points) 7. In 2020, the COVID-19 pandemic created a huge problem globally. More than 2 million people died from this disease as of Jan. 15, 2021. Some people say the elderly people are more likely to be infected (Claim A), and the survivors are mostly younger (Claim B). We would like to take a case from a country to see if these claims are reliable. City B had few hundreds of COVID-19 infected cases in 2020, and we took the COVID-19 cases in Hospital C in this city for the calculation. These are the figures of the confirmed infected cases in C hospital: Survival cases' age: 65, 88, 23, 8, 45, 55, 32, 22, 42, 65, 51, 15, 27, 65, 34, 13, 25, 31, 58, 32, 45, 67, 30, 58, 74 Confirmed death cases' age: 90, 18, 43, 67, 88, 89, 28, 55, 32, 66, 42, 49, 76, 59, 63, 2, 60, 84 The questions are to follow: a. First of all, from the age figures of the infected cases above, can you tell us if the infected cases' age is normally or skewed distributed (please show the calculations)? (5 points)

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