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A certain country has four regions: North, East, South, and West. The population 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.

a) What is the average size of a city in a region? (This is the arithmetic mean of the population of the cities, and is also the expected value of the population of a city chosen uniformly at random.)

b) 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?

  • For part A, letting X = the size of the city, I got E(X) = 1/10[3,000,000 + 4,000,000 + 5,000,000 + 8,000,000] = 2,000,000
  • For part B, 1/4[750,000 + 4/3(1,000,000) + 2,500,000 + 8,000,000] = about 3145833.33
  • My answers seem off but I am not sure where exactly in my work I am off. Any help would be greatly appreciated!
user102156
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    In part $(a)$ you are asked to calculate the average size of a city in a region, so the question probably wants you to calculate average sizes for each region separately (if not, then your answer gives the correct average size of a city in the COUNTRY). Your answer to part B is correct. – Shubham Johri Oct 21 '20 at 17:10
  • @ShubhamJohri can you please expand on your explanation on part A? What would the calculations be exactly if the question wanted me to calculate average sizes for each region separately? – user102156 Oct 21 '20 at 22:11
  • For calculating the average size of a city in a region, you restrict yourself to the population and number of cities in that region. For example, the average size of a city in North region is $3\text{ mil}/4=0.75\text{ mil}$. Similarly for other regions. – Shubham Johri Oct 21 '20 at 22:13
  • @ShubhamJohri I am not sure if I understood you correctly, but do mean: E(north) = 3/4, E(east) = 4/3, E(south) = 5/2, E(west) = 8/1? – user102156 Oct 22 '20 at 03:01

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