Actuarial Outpost SOA Sample Question #309
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#1
07-15-2018, 07:58 AM
 Mitsu96 Member Non-Actuary Join Date: Jan 2017 Location: Pennsylvania Studying for IFM Posts: 69
SOA Sample Question #309

Hello everyone,

I am struggling with this question (SOA 309):

The solution provided by SOA states that X and Y are the two coordinates of a resident. Thus, the distance to the origin is X + Y, following which the E[X + Y] is calculated. While, I understand how to calculate E[X + Y], I can't understand why or how X and Y are the coordinates of a resident and the distance to the origin is X + Y??
#2
07-15-2018, 08:31 AM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,179

You need to be more specific about what you don't understand by saying what you do understand or what conclusions you are drawing from the given information.

"Two sides of the square (of side 1) are on the positive axes". From that, you should conclude that coordinates of the square's corner on the x-axis are (x,0) and (x+1,0) for some value of x. Since it is a square (of side 1), the other two coordinates must be at (x,1) and (x+1,1) [or conceivably (x, -1) and (x+1, -1)]

Similarly, you should conclude that coordinates of the square on the y-axis are (0, y) and (0,y+1) for some value of y). Since it is a square (of side 1), the other two coordinates must be at (1,y) and (1,y+1) [or conceivably (-1, y) and (-1, y+1)]

But there are only 4 corners, so both descriptions must match the same 4 corners. Can you come up with any combination that works besides (0,0), (0,1), (1,0), (1,1)?

[You should reach that conclusion much easier from the density function, which is clearly nonzero only on that particular square, plus the location of the city hall, which presumably is going to be within the square. You might have expected that the city hall would be at the center, but the density function makes that unreasonable.]

If he can travel only on segments parallel to the axes, the it should be obvious that the shortest travel distance from (x,y) to (0,0) is x+y. [It would be pretty far-fetched to argue that an actual resident might not take the shortest route; for example going from (x,y) to (x,1), then to (0,1), then to (0,0).] There are many variations of the shortest route, for example (x,y) to (.5x,y) to (.5x,.3y) to (0,.3y), to (0,0), but all those variations have length x=y.
#3
07-15-2018, 10:45 AM
 Michael Mastroianni Member SOA Join Date: Jan 2018 Posts: 35

Perhaps this picture will help:
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Michael Mastroianni, ASA
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#4
07-15-2018, 11:07 AM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,578

This distance measure is sometimes referred to as the Manhattan metric. If you are on a grid and can only move parallel to the axes your distance from a point (x,y), to a point (xo,yo) is |x - xo| + |y-yo|. The technical name is the L1 norm while the traditional straight line distance where you square the differences and take the square root is the Euclidian distance or L2 norm.

 Tags expectation, joint distribution