Topic : Expectation A certain country has four regions: North, East, South, and
ID: 3125622 • Letter: T
Question
Topic : Expectation
A certain country has four regions: North, East, South, and West. The populations of these regions are 3 million, 4 million, 5 million, and 8 million, respectively. There are 4 cities in the North, 3 in the East, 2 in the South, and there is only 1 city in the West. Each person in the country lives in exactly one of these cities. What is the average size of a city in the country? (This is the arithmetic mean of the populations of the cities, and is also the expected value of the population of a city chosen uniformly at random.) Show that without further information it is impossible to find the variance of the population of a city chosen uniformly at random. That is, the variance depends on how the people within each region are allocated between the cities in that region. A region of the country is chosen uniformly at random, and then a city within that region is chosen uniformly at random. What is the expected population size of this randomly chosen city? Explain intuitively why the answer to (c) is larger than the answer to (a).Explanation / Answer
a) Total population in the country is (3+4+5+8)=20 million.
Total no of cities in the country (4+3+2+1)=10 million
The avg size of each city in the country (20/10) = 2 million
b) The information in not sufficient for calculating the variance of the population of the city.
Beacause the info provided here, each person in the country lives in exactly one of these cities.
It does not tell us how the people within each region are allocated between the cities in that region.
c) Cant understand the rest of the part